Abstract group $C_6\times D_4$

• Label: 48.45
• Order: \( 2^{4} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $9$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_6\times D_4$
Order:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) (generators)
Outer automorphisms:	$C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \)
Composition factors:	$C_2$ x 4, $C_3$
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Group statistics

Order	1	2	3	4	6	12
Elements	1	11	2	4	22	8	48
Conjugacy classes	1	7	2	2	14	4	30
Divisions	1	7	1	2	7	2	20
Autjugacy classes	1	3	1	1	3	1	10

Dimension	1	2	4
Irr. complex chars.	24	6	0	30
Irr. rational chars.	8	10	2	20

Minimal Presentations

Permutation degree:	$9$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$273$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	4	4

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{12}=[a,b]=[a,c]=1, c^{b}=c^{7} \rangle$
Permutation group:	$\langle(1,2)(3,4), (2,4)(5,6), (1,3)(2,4)(5,6), (7,9,8), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrr} -1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 3 & 3 & 6 \\ 2 & 4 & 3 \\ 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrr} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrr} 3 & 2 & 6 \\ 2 & 4 & 3 \\ 0 & 0 & 6 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{7})$
Transitive group:	24T38				more information
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $D_4$
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $C_6$ (2)	$C_{12}$ $\,\rtimes\,$ $C_2^2$ (2)	$(C_2\times C_4)$ $\,\rtimes\,$ $C_6$	$C_4$ $\,\rtimes\,$ $(C_2\times C_6)$ (2)	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $C_2^3$	$(C_2\times C_6)$ . $C_2^2$	$C_2^2$ . $(C_2\times C_6)$	$C_2$ . $(C_2^2\times C_6)$	more information
Aut. group:	$\Aut(C_2\times C_{28})$	$\Aut(C_7\times D_4)$	$\Aut(C_2\times C_{36})$	$\Aut(D_4\times C_9)$

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

Homology

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

Subgroups

There are 70 subgroups in 54 conjugacy classes, 38 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_6$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_6\times D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 48: $C_6\times D_4$

Order 24: $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$

Order 16: $C_2\times D_4$

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

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 48: $C_6\times D_4$

Order 24: $C_2\times C_{12}$, $C_2^2\times C_6$, $C_3\times D_4$

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 3, $C_{12}$

Order 8: $C_2^3$, $D_4$, $C_2\times C_4$

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 3, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 48: $C_6\times D_4$ ($C_1$)

Order 24: $C_3\times D_4$ ($C_2$) x 4, $C_2^2\times C_6$ ($C_2$) x 2, $C_2\times C_{12}$ ($C_2$)

Order 16: $C_2\times D_4$ ($C_3$)

Order 12: $C_2\times C_6$ ($C_2^2$) x 5, $C_{12}$ ($C_2^2$) x 2

Order 8: $D_4$ ($C_6$) x 4, $C_2^3$ ($C_6$) x 2, $C_2\times C_4$ ($C_6$)

Order 6: $C_6$ ($D_4$) x 2, $C_6$ ($C_2^3$)

Order 4: $C_2^2$ ($C_2\times C_6$) x 5, $C_4$ ($C_2\times C_6$) x 2

Order 3: $C_3$ ($C_2\times D_4$)

Order 2: $C_2$ ($C_3\times D_4$) x 2, $C_2$ ($C_2^2\times C_6$)

Order 1: $C_1$ ($C_6\times D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 48: $C_6\times D_4$ ($C_1$)

Order 24: $C_2\times C_{12}$ ($C_2$), $C_2^2\times C_6$ ($C_2$), $C_3\times D_4$ ($C_2$)

Order 16: $C_2\times D_4$ ($C_3$)

Order 12: $C_2\times C_6$ ($C_2^2$) x 2, $C_{12}$ ($C_2^2$)

Order 8: $C_2^3$ ($C_6$), $D_4$ ($C_6$), $C_2\times C_4$ ($C_6$)

Order 6: $C_6$ ($C_2^3$), $C_6$ ($D_4$)

Order 4: $C_2^2$ ($C_2\times C_6$) x 2, $C_4$ ($C_2\times C_6$)

Order 3: $C_3$ ($C_2\times D_4$)

Order 2: $C_2$ ($C_2^2\times C_6$), $C_2$ ($C_3\times D_4$)

Order 1: $C_1$ ($C_6\times D_4$)

Series

Derived series	$C_6\times D_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Chief series	$C_6\times D_4$	$\rhd$	$C_2^2\times C_6$	$\rhd$	$C_2\times C_6$	$\rhd$	$C_6$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_6\times D_4$	$\rhd$	$C_2$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_2\times C_6$	$\lhd$	$C_6\times D_4$

Supergroups

This group is a maximal subgroup of 111 larger groups in the database.

This group is a maximal quotient of 107 larger groups in the database.

Character theory

Complex character table

See the $30 \times 30$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.