Abstract group $C_6\times \PGL(2,7)$

• Label: 2016.b
• Order: \( 2^{5} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{6} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $13$
• Trans deg.: $48$
• Rank: $2$

Group information

Description:	$C_6\times \PGL(2,7)$
Order:	\(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^2\times \PGL(2,7)$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Outer automorphisms:	$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
Composition factors:	$C_2$ x 2, $C_3$, $\GL(3,2)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	8	12	14	21	24	42
Elements	1	99	170	84	702	48	168	168	48	96	336	96	2016
Conjugacy classes	1	5	5	2	19	1	4	4	1	2	8	2	54
Divisions	1	5	3	2	11	1	2	2	1	1	2	1	32
Autjugacy classes	1	4	3	2	8	1	2	2	1	1	2	1	28

Dimension	1	2	6	7	8	12	14	16	24
Irr. complex chars.	12	0	18	12	12	0	0	0	0	54
Irr. rational chars.	4	4	2	4	4	4	4	4	2	32

Minimal Presentations

Permutation degree:	$13$
Transitive degree:	$48$
Rank:	$2$
Inequivalent generating pairs:	$828$

Minimal degrees of faithful linear representations

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

Constructions

Groups of Lie type:	$\CO(3,7)$
Permutation group:	Degree $13$ $\langle(9,10,11)(12,13), (1,2,4,5,3,6), (2,4,5)(6,7,8), (1,3)(2,5)(7,8)(12,13)\rangle$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $\PGL(2,7)$
Semidirect product:	$(C_6\times \GL(3,2))$ $\,\rtimes\,$ $C_2$	$(C_2\times \GL(3,2))$ $\,\rtimes\,$ $C_6$	$\GL(3,2)$ $\,\rtimes\,$ $(C_2\times C_6)$	$(C_3\times \GL(3,2))$ $\,\rtimes\,$ $C_2^2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_7\times \GL(3,2))$	$\Aut(C_9\times \GL(3,2))$	$\Aut(C_7\times \SL(2,7))$	$\Aut(C_7\times \PGL(2,7))$	all 7

Elements of the group are displayed as matrices in $\CO(3,7)$.

Homology

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

Subgroups

There are 3362 subgroups in 157 conjugacy classes, 14 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $\PGL(2,7)$
Commutator:	$G' \simeq$ $\GL(3,2)$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_6\times \PGL(2,7)$
Fitting:	$\operatorname{Fit} \simeq$ $C_6$	$G/\operatorname{Fit} \simeq$ $\PGL(2,7)$
Radical:	$R \simeq$ $C_6$	$G/R \simeq$ $\PGL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_6\times \GL(3,2)$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_8$
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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Classes of subgroups up to conjugation

Order 2016: $C_6\times \PGL(2,7)$

Order 1008: $C_3\times \PGL(2,7)$ x 2, $C_6\times \GL(3,2)$

Order 672: $C_2\times \PGL(2,7)$

Order 504: $C_3\times \GL(3,2)$

Order 336: $\PGL(2,7)$ x 2, $C_2\times \GL(3,2)$

Order 252: $C_6\times F_7$

Order 168: $\GL(3,2)$

Order 144: $C_6\times S_4$

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

Order 96: $C_6\times D_8$

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

Order 72: $C_3\times S_4$ x 2, $C_6\times D_6$, $C_6\times A_4$

Order 63: $C_{21}:C_3$

Order 48: $C_3\times D_8$ x 4, $C_6\times D_4$ x 2, $C_2\times S_4$, $C_2\times C_{24}$

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

Order 36: $C_6\times S_3$ x 6, $C_6^2$, $C_3\times A_4$

Order 32: $C_2\times D_8$

Order 28: $D_{14}$

Order 24: $C_3\times D_4$ x 6, $C_{24}$ x 2, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2, $S_4$ x 2, $C_2\times C_{12}$, $C_2\times D_6$

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

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $D_8$ x 4, $C_2\times D_4$ x 2, $C_2\times C_8$

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

Order 12: $C_2\times C_6$ x 9, $D_6$ x 6, $A_4$ x 2, $C_{12}$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 6, $C_2^3$ x 2, $C_8$ x 2, $C_2\times C_4$

Order 7: $C_7$

Order 6: $C_6$ x 11, $S_3$ x 4

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2016: $C_6\times \PGL(2,7)$

Order 1008: $C_6\times \GL(3,2)$, $C_3\times \PGL(2,7)$

Order 672: $C_2\times \PGL(2,7)$

Order 504: $C_3\times \GL(3,2)$

Order 336: $C_2\times \GL(3,2)$, $\PGL(2,7)$

Order 252: $C_6\times F_7$

Order 168: $\GL(3,2)$

Order 144: $C_6\times S_4$

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

Order 96: $C_6\times D_8$

Order 84: $C_2\times F_7$ x 2, $C_3\times D_{14}$

Order 72: $C_3\times S_4$ x 2, $C_6\times D_6$, $C_6\times A_4$

Order 63: $C_{21}:C_3$

Order 48: $C_6\times D_4$ x 2, $C_3\times D_8$ x 2, $C_2\times S_4$, $C_2\times C_{24}$

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

Order 36: $C_6\times S_3$ x 4, $C_6^2$, $C_3\times A_4$

Order 32: $C_2\times D_8$

Order 28: $D_{14}$

Order 24: $C_3\times D_4$ x 5, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2, $S_4$ x 2, $C_2\times C_{12}$, $C_{24}$, $C_2\times D_6$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 18: $C_3\times S_3$ x 3, $C_3\times C_6$ x 2

Order 16: $D_8$ x 2, $C_2\times D_4$ x 2, $C_2\times C_8$

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

Order 12: $C_2\times C_6$ x 8, $D_6$ x 4, $A_4$ x 2, $C_{12}$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 5, $C_2^3$ x 2, $C_2\times C_4$, $C_8$

Order 7: $C_7$

Order 6: $C_6$ x 8, $S_3$ x 3

Order 4: $C_2^2$ x 6, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2016: $C_6\times \PGL(2,7)$ ($C_1$)

Order 1008: $C_3\times \PGL(2,7)$ ($C_2$) x 2, $C_6\times \GL(3,2)$ ($C_2$)

Order 672: $C_2\times \PGL(2,7)$ ($C_3$)

Order 504: $C_3\times \GL(3,2)$ ($C_2^2$)

Order 336: $\PGL(2,7)$ ($C_6$) x 2, $C_2\times \GL(3,2)$ ($C_6$)

Order 168: $\GL(3,2)$ ($C_2\times C_6$)

Order 6: $C_6$ ($\PGL(2,7)$)

Order 3: $C_3$ ($C_2\times \PGL(2,7)$)

Order 2: $C_2$ ($C_3\times \PGL(2,7)$)

Order 1: $C_1$ ($C_6\times \PGL(2,7)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2016: $C_6\times \PGL(2,7)$ ($C_1$)

Order 1008: $C_6\times \GL(3,2)$ ($C_2$), $C_3\times \PGL(2,7)$ ($C_2$)

Order 672: $C_2\times \PGL(2,7)$ ($C_3$)

Order 504: $C_3\times \GL(3,2)$ ($C_2^2$)

Order 336: $C_2\times \GL(3,2)$ ($C_6$), $\PGL(2,7)$ ($C_6$)

Order 168: $\GL(3,2)$ ($C_2\times C_6$)

Order 6: $C_6$ ($\PGL(2,7)$)

Order 3: $C_3$ ($C_2\times \PGL(2,7)$)

Order 2: $C_2$ ($C_3\times \PGL(2,7)$)

Order 1: $C_1$ ($C_6\times \PGL(2,7)$)

Series

Derived series	$C_6\times \PGL(2,7)$	$\rhd$	$\GL(3,2)$
Chief series	$C_6\times \PGL(2,7)$	$\rhd$	$C_6\times \GL(3,2)$	$\rhd$	$C_6$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_6\times \PGL(2,7)$	$\rhd$	$\GL(3,2)$
Upper central series	$C_1$	$\lhd$	$C_6$

Supergroups

This group is a maximal subgroup of 12 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

See the $54 \times 54$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.