Abstract group $F_7\times C_{48}$

• Label: 2016.y
• Order: \( 2^{5} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{6} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $26$
• Trans deg.: $336$
• Rank: $2$

Group information

Description:	$F_7\times C_{48}$
Order:	\(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \)
Exponent:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Automorphism group:	Group of order \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \) (generators)
Outer automorphisms:	$C_{12}:C_2^3$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2, $C_7$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

Group statistics

Order	1	2	3	4	6	7	8	12	14	16	21	24	28	42	48	56	84	112	168	336
Elements	1	15	44	16	156	6	32	200	6	64	12	400	12	12	800	24	24	48	48	96	2016
Conjugacy classes	1	3	8	4	24	1	8	32	1	16	2	64	2	2	128	4	4	8	8	16	336
Divisions	1	3	4	2	12	1	2	8	1	2	1	8	1	1	8	1	1	1	1	1	60
Autjugacy classes	1	2	3	2	6	1	2	6	1	2	1	6	1	1	6	1	1	1	1	1	46

Dimension	1	2	4	6	8	12	16	24	48	96
Irr. complex chars.	288	0	0	48	0	0	0	0	0	0	336
Irr. rational chars.	4	18	10	2	10	3	8	2	2	1	60

Minimal Presentations

Permutation degree:	$26$
Transitive degree:	$336$
Rank:	$2$
Inequivalent generating pairs:	$192$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	96
Arbitrary	6	8	16

Constructions

Presentation:	$\langle a, b \mid b^{336}=1, a^{6}=b^{126}, b^{a}=b^{241} \rangle$
Permutation group:	Degree $26$ $\langle(1,2,5,11,7,8,12,15)(3,9,13,16,6,4,10,14)(17,18,19), (1,3,2,9,5,13,11,16,7,6,8,4,12,10,15,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 4 & 3 \\ 6 & 3 & 1 & 4 \\ 4 & 5 & 2 & 5 \\ 2 & 4 & 1 & 5 \end{array}\right), \left(\begin{array}{rrrr} 6 & 4 & 1 & 2 \\ 5 & 0 & 1 & 1 \\ 1 & 6 & 1 & 3 \\ 2 & 1 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 6 & 1 & 1 & 5 \\ 6 & 5 & 1 & 1 \\ 0 & 4 & 2 & 5 \\ 6 & 4 & 6 & 6 \end{array}\right), \left(\begin{array}{rrrr} 0 & 6 & 5 & 3 \\ 4 & 5 & 5 & 5 \\ 5 & 2 & 3 & 1 \\ 3 & 5 & 3 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 2 & 4 \\ 1 & 5 & 4 & 2 \\ 6 & 2 & 0 & 1 \\ 2 & 2 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 5 & 1 & 3 \\ 1 & 3 & 1 & 4 \\ 4 & 4 & 0 & 1 \\ 5 & 6 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	$C_{16}$ $\, \times\, $ $C_3$ $\, \times\, $ $F_7$
Semidirect product:	$C_{336}$ $\,\rtimes\,$ $C_6$	$(D_7\times C_{48})$ $\,\rtimes\,$ $C_3$	$(C_3\times D_7)$ $\,\rtimes\,$ $C_{48}$ (2)	$D_7$ $\,\rtimes\,$ $(C_3\times C_{48})$ (2)	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{56}$ . $C_6^2$	$(C_8\times F_7)$ . $C_6$ (3)	$(C_6\times F_7)$ . $C_8$	$C_8$ . $(C_6\times F_7)$	all 38
Aut. group:	$\Aut(C_7:C_{153})$

Elements of the group are displayed as words in the generators from the presentation given above.

Homology

Abelianization:	$C_{6} \times C_{48} \simeq C_{2} \times C_{16} \times C_{3}^{2}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 612 subgroups in 168 conjugacy classes, 94 normal (46 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{48}$	$G/Z \simeq$ $F_7$
Commutator:	$G' \simeq$ $C_7$	$G/G' \simeq$ $C_6\times C_{48}$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_6\times F_7$
Fitting:	$\operatorname{Fit} \simeq$ $C_{336}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $F_7\times C_{48}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{24}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 2016: $F_7\times C_{48}$

Order 1008: $C_{24}\times F_7$, $C_{21}:C_{48}$, $C_{21}:C_{48}$

Order 672: $C_{16}\times F_7$ x 3, $D_7\times C_{48}$

Order 504: $C_{12}\times F_7$, $C_{21}:C_{24}$, $C_{21}:C_{24}$

Order 336: $C_8\times F_7$ x 3, $C_7:C_{48}$ x 3, $C_7:C_{48}$ x 3, $C_{336}$, $D_7\times C_{24}$, $C_7:C_{48}$

Order 288: $C_6\times C_{48}$

Order 252: $C_6\times F_7$, $C_{21}:C_{12}$, $C_{21}:C_{12}$

Order 224: $D_7\times C_{16}$

Order 168: $C_4\times F_7$ x 3, $C_7:C_{24}$ x 3, $C_7:C_{24}$ x 3, $C_{168}$, $C_7:C_{24}$, $C_{12}\times D_7$

Order 144: $C_3\times C_{48}$ x 2, $C_6\times C_{24}$

Order 126: $C_3\times F_7$ x 2, $C_{21}:C_6$

Order 112: $C_8\times D_7$, $C_{112}$, $C_7:C_{16}$

Order 96: $C_2\times C_{48}$ x 4

Order 84: $C_2\times F_7$ x 3, $C_7:C_{12}$ x 3, $C_7:C_{12}$ x 3, $C_{84}$, $C_7:C_{12}$, $C_3\times D_{14}$

Order 72: $C_3\times C_{24}$ x 2, $C_6\times C_{12}$

Order 63: $C_{21}:C_3$

Order 56: $C_4\times D_7$, $C_{56}$, $C_7:C_8$

Order 48: $C_{48}$ x 8, $C_2\times C_{24}$ x 4

Order 42: $F_7$ x 6, $C_7:C_6$ x 3, $C_3\times D_7$ x 2, $C_{42}$

Order 36: $C_3\times C_{12}$ x 2, $C_6^2$

Order 32: $C_2\times C_{16}$

Order 28: $D_{14}$, $C_{28}$, $C_7:C_4$

Order 24: $C_{24}$ x 8, $C_2\times C_{12}$ x 4

Order 21: $C_7:C_3$ x 3, $C_{21}$

Order 18: $C_3\times C_6$ x 3

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 14: $D_7$ x 2, $C_{14}$

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 4

Order 9: $C_3^2$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 7: $C_7$

Order 6: $C_6$ x 12

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2016: $F_7\times C_{48}$

Order 1008: $C_{24}\times F_7$, $C_{21}:C_{48}$, $C_{21}:C_{48}$

Order 672: $C_{16}\times F_7$, $D_7\times C_{48}$

Order 504: $C_{12}\times F_7$, $C_{21}:C_{24}$, $C_{21}:C_{24}$

Order 336: $C_8\times F_7$, $C_{336}$, $D_7\times C_{24}$, $C_7:C_{48}$, $C_7:C_{48}$, $C_7:C_{48}$

Order 288: $C_6\times C_{48}$

Order 252: $C_6\times F_7$, $C_{21}:C_{12}$, $C_{21}:C_{12}$

Order 224: $D_7\times C_{16}$

Order 168: $C_4\times F_7$, $C_{168}$, $C_7:C_{24}$, $C_{12}\times D_7$, $C_7:C_{24}$, $C_7:C_{24}$

Order 144: $C_3\times C_{48}$ x 2, $C_6\times C_{24}$

Order 126: $C_3\times F_7$, $C_{21}:C_6$

Order 112: $C_8\times D_7$, $C_{112}$, $C_7:C_{16}$

Order 96: $C_2\times C_{48}$ x 2

Order 84: $C_2\times F_7$, $C_{84}$, $C_7:C_{12}$, $C_7:C_{12}$, $C_3\times D_{14}$, $C_7:C_{12}$

Order 72: $C_3\times C_{24}$ x 2, $C_6\times C_{12}$

Order 63: $C_{21}:C_3$

Order 56: $C_4\times D_7$, $C_{56}$, $C_7:C_8$

Order 48: $C_{48}$ x 4, $C_2\times C_{24}$ x 2

Order 42: $C_{42}$, $C_3\times D_7$, $C_7:C_6$, $F_7$

Order 36: $C_3\times C_{12}$ x 2, $C_6^2$

Order 32: $C_2\times C_{16}$

Order 28: $D_{14}$, $C_{28}$, $C_7:C_4$

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$ x 2

Order 21: $C_{21}$, $C_7:C_3$

Order 18: $C_3\times C_6$ x 2

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 14: $C_{14}$, $D_7$

Order 12: $C_{12}$ x 4, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 7: $C_7$

Order 6: $C_6$ x 4

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2016: $F_7\times C_{48}$ ($C_1$)

Order 1008: $C_{24}\times F_7$ ($C_2$), $C_{21}:C_{48}$ ($C_2$), $C_{21}:C_{48}$ ($C_2$)

Order 672: $C_{16}\times F_7$ ($C_3$) x 3, $D_7\times C_{48}$ ($C_3$)

Order 504: $C_{12}\times F_7$ ($C_4$), $C_{21}:C_{24}$ ($C_2^2$), $C_{21}:C_{24}$ ($C_4$)

Order 336: $C_8\times F_7$ ($C_6$) x 3, $C_7:C_{48}$ ($C_6$) x 3, $C_7:C_{48}$ ($C_6$) x 3, $C_{336}$ ($C_6$), $D_7\times C_{24}$ ($C_6$), $C_7:C_{48}$ ($C_6$)

Order 252: $C_6\times F_7$ ($C_8$), $C_{21}:C_{12}$ ($C_2\times C_4$), $C_{21}:C_{12}$ ($C_8$)

Order 224: $D_7\times C_{16}$ ($C_3^2$)

Order 168: $C_4\times F_7$ ($C_{12}$) x 3, $C_7:C_{24}$ ($C_2\times C_6$) x 3, $C_7:C_{24}$ ($C_{12}$) x 3, $C_{168}$ ($C_2\times C_6$), $C_7:C_{24}$ ($C_{12}$), $C_{12}\times D_7$ ($C_{12}$)

Order 126: $C_3\times F_7$ ($C_{16}$) x 2, $C_{21}:C_6$ ($C_2\times C_8$)

Order 112: $C_8\times D_7$ ($C_3\times C_6$), $C_{112}$ ($C_3\times C_6$), $C_7:C_{16}$ ($C_3\times C_6$)

Order 84: $C_2\times F_7$ ($C_{24}$) x 3, $C_7:C_{12}$ ($C_2\times C_{12}$) x 3, $C_7:C_{12}$ ($C_{24}$) x 3, $C_{84}$ ($C_2\times C_{12}$), $C_7:C_{12}$ ($C_{24}$), $C_3\times D_{14}$ ($C_{24}$)

Order 63: $C_{21}:C_3$ ($C_2\times C_{16}$)

Order 56: $C_4\times D_7$ ($C_3\times C_{12}$), $C_{56}$ ($C_6^2$), $C_7:C_8$ ($C_3\times C_{12}$)

Order 48: $C_{48}$ ($F_7$)

Order 42: $F_7$ ($C_{48}$) x 6, $C_7:C_6$ ($C_2\times C_{24}$) x 3, $C_3\times D_7$ ($C_{48}$) x 2, $C_{42}$ ($C_2\times C_{24}$)

Order 28: $D_{14}$ ($C_3\times C_{24}$), $C_{28}$ ($C_6\times C_{12}$), $C_7:C_4$ ($C_3\times C_{24}$)

Order 24: $C_{24}$ ($C_2\times F_7$)

Order 21: $C_7:C_3$ ($C_2\times C_{48}$) x 3, $C_{21}$ ($C_2\times C_{48}$)

Order 16: $C_{16}$ ($C_3\times F_7$)

Order 14: $D_7$ ($C_3\times C_{48}$) x 2, $C_{14}$ ($C_6\times C_{24}$)

Order 12: $C_{12}$ ($C_4\times F_7$)

Order 8: $C_8$ ($C_6\times F_7$)

Order 7: $C_7$ ($C_6\times C_{48}$)

Order 6: $C_6$ ($C_8\times F_7$)

Order 4: $C_4$ ($C_{12}\times F_7$)

Order 3: $C_3$ ($C_{16}\times F_7$)

Order 2: $C_2$ ($C_{24}\times F_7$)

Order 1: $C_1$ ($F_7\times C_{48}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2016: $F_7\times C_{48}$ ($C_1$)

Order 1008: $C_{24}\times F_7$ ($C_2$), $C_{21}:C_{48}$ ($C_2$), $C_{21}:C_{48}$ ($C_2$)

Order 672: $C_{16}\times F_7$ ($C_3$), $D_7\times C_{48}$ ($C_3$)

Order 504: $C_{12}\times F_7$ ($C_4$), $C_{21}:C_{24}$ ($C_2^2$), $C_{21}:C_{24}$ ($C_4$)

Order 336: $C_8\times F_7$ ($C_6$), $C_{336}$ ($C_6$), $D_7\times C_{24}$ ($C_6$), $C_7:C_{48}$ ($C_6$), $C_7:C_{48}$ ($C_6$), $C_7:C_{48}$ ($C_6$)

Order 252: $C_6\times F_7$ ($C_8$), $C_{21}:C_{12}$ ($C_2\times C_4$), $C_{21}:C_{12}$ ($C_8$)

Order 224: $D_7\times C_{16}$ ($C_3^2$)

Order 168: $C_4\times F_7$ ($C_{12}$), $C_{168}$ ($C_2\times C_6$), $C_7:C_{24}$ ($C_{12}$), $C_{12}\times D_7$ ($C_{12}$), $C_7:C_{24}$ ($C_2\times C_6$), $C_7:C_{24}$ ($C_{12}$)

Order 126: $C_3\times F_7$ ($C_{16}$), $C_{21}:C_6$ ($C_2\times C_8$)

Order 112: $C_8\times D_7$ ($C_3\times C_6$), $C_{112}$ ($C_3\times C_6$), $C_7:C_{16}$ ($C_3\times C_6$)

Order 84: $C_2\times F_7$ ($C_{24}$), $C_{84}$ ($C_2\times C_{12}$), $C_7:C_{12}$ ($C_{24}$), $C_7:C_{12}$ ($C_2\times C_{12}$), $C_3\times D_{14}$ ($C_{24}$), $C_7:C_{12}$ ($C_{24}$)

Order 63: $C_{21}:C_3$ ($C_2\times C_{16}$)

Order 56: $C_4\times D_7$ ($C_3\times C_{12}$), $C_{56}$ ($C_6^2$), $C_7:C_8$ ($C_3\times C_{12}$)

Order 48: $C_{48}$ ($F_7$)

Order 42: $C_{42}$ ($C_2\times C_{24}$), $C_3\times D_7$ ($C_{48}$), $C_7:C_6$ ($C_2\times C_{24}$), $F_7$ ($C_{48}$)

Order 28: $D_{14}$ ($C_3\times C_{24}$), $C_{28}$ ($C_6\times C_{12}$), $C_7:C_4$ ($C_3\times C_{24}$)

Order 24: $C_{24}$ ($C_2\times F_7$)

Order 21: $C_{21}$ ($C_2\times C_{48}$), $C_7:C_3$ ($C_2\times C_{48}$)

Order 16: $C_{16}$ ($C_3\times F_7$)

Order 14: $C_{14}$ ($C_6\times C_{24}$), $D_7$ ($C_3\times C_{48}$)

Order 12: $C_{12}$ ($C_4\times F_7$)

Order 8: $C_8$ ($C_6\times F_7$)

Order 7: $C_7$ ($C_6\times C_{48}$)

Order 6: $C_6$ ($C_8\times F_7$)

Order 4: $C_4$ ($C_{12}\times F_7$)

Order 3: $C_3$ ($C_{16}\times F_7$)

Order 2: $C_2$ ($C_{24}\times F_7$)

Order 1: $C_1$ ($F_7\times C_{48}$)

Series

Derived series

$F_7\times C_{48}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Chief series

$F_7\times C_{48}$

$\rhd$

$C_{21}:C_{48}$

$\rhd$

$C_{336}$

$\rhd$

$C_{168}$

$\rhd$

$C_{84}$

$\rhd$

$C_{42}$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$F_7\times C_{48}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_{48}$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $336 \times 336$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $60 \times 60$ rational character table.