Abstract group 576.5330: $(C_6\times C_{12}).D_4$

• Label: 576.5330
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $26$
• Trans deg.: $192$
• Rank: $3$

Group information

Description:	$(C_6\times C_{12}).D_4$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3^2.C_2^6.C_2^4$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	3	8	172	24	144	224	576
Conjugacy classes	1	3	2	11	6	4	12	39
Divisions	1	3	2	10	6	2	8	32
Autjugacy classes	1	3	2	7	6	1	5	25

Dimension	1	2	4	8
Irr. complex chars.	8	10	17	4	39
Irr. rational chars.	8	4	12	8	32

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	$6720$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	10	14

Constructions

Presentation:	${\langle a, b, c, d \mid c^{12}=d^{6}=[a,d]=[c,d]=1, a^{2}=c^{6}d^{3}, b^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $26$ $\langle(4,6)(7,8,11,14)(9,15,17,22)(10,18,20,13)(12,19,21,16), (1,2)(3,4)(5,6) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^2:C_4)$ $\,\rtimes\,$ $Q_{16}$ (2)	$C_3^2$ $\,\rtimes\,$ $(C_4:Q_{16})$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6.D_6)$ . $D_4$	$(C_6\times C_{12})$ . $D_4$	$C_2$ . $(S_3^2:D_4)$	$(C_3^2:Q_8)$ . $D_4$ (2)	all 24

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$2$

Subgroups

There are 978 subgroups in 177 conjugacy classes, 35 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_3^2:C_2^2$
Commutator:	$G' \simeq$ $C_6.D_6$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $S_3^2:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{12}$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $(C_6\times C_{12}).D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4:Q_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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 576: $(C_6\times C_{12}).D_4$

Order 288: $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $(C_6\times C_{12}).C_4$

Order 144: $C_3^2:Q_{16}$ x 4, $C_2^2.S_3^2$ x 3, $C_2^2.S_3^2$ x 2, $C_6^2.C_4$ x 2, $C_3^2:C_4^2$, $C_{12}:C_{12}$, $C_6^2.C_2^2$, $C_3^2:C_4^2$

Order 96: $C_{12}:Q_8$, $C_3:(C_4\times Q_8)$

Order 72: $C_3^2:Q_8$ x 7, $C_6:C_{12}$ x 4, $C_6.D_6$ x 2, $C_3^2:C_8$ x 2, $C_6\times C_{12}$

Order 64: $C_4:Q_{16}$

Order 48: $C_6.D_4$ x 4, $C_{12}:C_4$ x 4, $C_6:Q_8$ x 3, $C_4:C_{12}$ x 2, $C_{12}:C_4$

Order 36: $C_3:C_{12}$ x 5, $C_3^2:C_4$ x 4, $C_3\times C_{12}$, $C_6^2$

Order 32: $C_2\times Q_{16}$ x 2, $Q_8:C_4$ x 2, $C_4:Q_8$, $C_4\times Q_8$, $C_4:C_8$

Order 24: $C_6:C_4$ x 8, $C_3:Q_8$ x 8, $C_2\times C_{12}$ x 6

Order 18: $C_3\times C_6$ x 3

Order 16: $Q_{16}$ x 4, $C_4:C_4$ x 4, $C_2\times Q_8$ x 3, $C_2\times C_8$ x 2, $C_4^2$ x 2

Order 12: $C_3:C_4$ x 13, $C_{12}$ x 8, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $Q_8$ x 7, $C_2\times C_4$ x 7, $C_8$ x 2

Order 6: $C_6$ x 6

Order 4: $C_4$ x 10, $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $(C_6\times C_{12}).D_4$

Order 288: $C_6^2.D_4$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $(C_6\times C_{12}).C_4$, $C_6^2.D_4$

Order 144: $C_2^2.S_3^2$ x 2, $C_2^2.S_3^2$ x 2, $C_3^2:C_4^2$, $C_{12}:C_{12}$, $C_6^2.C_2^2$, $C_3^2:C_4^2$, $C_6^2.C_4$, $C_3^2:Q_{16}$

Order 96: $C_{12}:Q_8$, $C_3:(C_4\times Q_8)$

Order 72: $C_3^2:Q_8$ x 4, $C_6:C_{12}$ x 3, $C_6.D_6$ x 2, $C_6\times C_{12}$, $C_3^2:C_8$

Order 64: $C_4:Q_{16}$

Order 48: $C_{12}:C_4$ x 4, $C_6.D_4$ x 3, $C_6:Q_8$ x 2, $C_4:C_{12}$ x 2, $C_{12}:C_4$

Order 36: $C_3^2:C_4$ x 3, $C_3:C_{12}$ x 3, $C_3\times C_{12}$, $C_6^2$

Order 32: $C_2\times Q_{16}$, $C_4:Q_8$, $C_4\times Q_8$, $C_4:C_8$, $Q_8:C_4$

Order 24: $C_6:C_4$ x 7, $C_2\times C_{12}$ x 5, $C_3:Q_8$ x 4

Order 18: $C_3\times C_6$ x 3

Order 16: $C_4:C_4$ x 4, $C_4^2$ x 2, $C_2\times Q_8$ x 2, $Q_{16}$, $C_2\times C_8$

Order 12: $C_3:C_4$ x 9, $C_{12}$ x 5, $C_2\times C_6$ x 2

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 6, $Q_8$ x 4, $C_8$

Order 6: $C_6$ x 6

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $(C_6\times C_{12}).D_4$ ($C_1$)

Order 288: $C_6^2.D_4$ ($C_2$) x 2, $C_6^2.D_4$ ($C_2$) x 2, $C_6^2.C_2^3$ ($C_2$), $C_6^2.C_2^3$ ($C_2$), $(C_6\times C_{12}).C_4$ ($C_2$)

Order 144: $C_2^2.S_3^2$ ($C_2^2$) x 3, $C_6^2.C_4$ ($C_2^2$) x 2, $C_3^2:C_4^2$ ($C_2^2$), $C_2^2.S_3^2$ ($C_2^2$)

Order 72: $C_3^2:Q_8$ ($D_4$) x 2, $C_6\times C_{12}$ ($D_4$), $C_6.D_6$ ($C_2^3$), $C_6.D_6$ ($D_4$)

Order 36: $C_3^2:C_4$ ($Q_{16}$) x 2, $C_3^2:C_4$ ($D_4:C_2$), $C_3^2:C_4$ ($C_2\times D_4$), $C_6^2$ ($C_2\times D_4$)

Order 18: $C_3\times C_6$ ($Q_{16}:C_2$), $C_3\times C_6$ ($C_2\times Q_{16}$), $C_3\times C_6$ ($C_4:D_4$)

Order 9: $C_3^2$ ($C_4:Q_{16}$)

Order 8: $C_2\times C_4$ ($\SOPlus(4,2)$)

Order 4: $C_2^2$ ($S_3^2:C_2^2$)

Order 2: $C_2$ ($C_6^2.D_4$), $C_2$ ($S_3^2:D_4$), $C_2$ ($C_3^2:(C_2\times Q_{16})$)

Order 1: $C_1$ ($(C_6\times C_{12}).D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $(C_6\times C_{12}).D_4$ ($C_1$)

Order 288: $C_6^2.D_4$ ($C_2$), $C_6^2.C_2^3$ ($C_2$), $C_6^2.C_2^3$ ($C_2$), $(C_6\times C_{12}).C_4$ ($C_2$), $C_6^2.D_4$ ($C_2$)

Order 144: $C_2^2.S_3^2$ ($C_2^2$) x 2, $C_3^2:C_4^2$ ($C_2^2$), $C_2^2.S_3^2$ ($C_2^2$), $C_6^2.C_4$ ($C_2^2$)

Order 72: $C_6\times C_{12}$ ($D_4$), $C_6.D_6$ ($C_2^3$), $C_6.D_6$ ($D_4$), $C_3^2:Q_8$ ($D_4$)

Order 36: $C_3^2:C_4$ ($Q_{16}$), $C_3^2:C_4$ ($D_4:C_2$), $C_3^2:C_4$ ($C_2\times D_4$), $C_6^2$ ($C_2\times D_4$)

Order 18: $C_3\times C_6$ ($Q_{16}:C_2$), $C_3\times C_6$ ($C_2\times Q_{16}$), $C_3\times C_6$ ($C_4:D_4$)

Order 9: $C_3^2$ ($C_4:Q_{16}$)

Order 8: $C_2\times C_4$ ($\SOPlus(4,2)$)

Order 4: $C_2^2$ ($S_3^2:C_2^2$)

Order 2: $C_2$ ($C_6^2.D_4$), $C_2$ ($S_3^2:D_4$), $C_2$ ($C_3^2:(C_2\times Q_{16})$)

Order 1: $C_1$ ($(C_6\times C_{12}).D_4$)

Series

Derived series

$(C_6\times C_{12}).D_4$

$\rhd$

$C_6.D_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$(C_6\times C_{12}).D_4$

$\rhd$

$C_6^2.D_4$

$\rhd$

$C_2^2.S_3^2$

$\rhd$

$C_6.D_6$

$\rhd$

$C_3^2:C_4$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$(C_6\times C_{12}).D_4$

$\rhd$

$C_6.D_6$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

See the $39 \times 39$ 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.