Abstract group 576.5410: $C_2^3.\SOPlus(4,2)$

• Label: 576.5410
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $14$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2^3.\SOPlus(4,2)$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_3^2.C_2^6.C_2^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
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	91	8	84	152	144	96	576
Conjugacy classes	1	7	2	6	14	2	4	36
Divisions	1	7	2	6	11	2	4	33
Autjugacy classes	1	6	2	3	10	1	2	25

Dimension	1	2	4	8	16
Irr. complex chars.	8	6	18	4	0	36
Irr. rational chars.	8	6	14	4	1	33

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$26880$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	8
Arbitrary	4	8	8

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{4}=d^{12}=e^{3}=[a,b]=[a,e]=[b,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(4,6)(7,8)(9,11)(10,13)(12,14), (1,2)(3,4)(5,6)(8,11)(9,14)(12,13), (9,14) \!\cdots\! \rangle$
Transitive group:	24T1375	24T1379	24T1425	24T1427	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(D_6:S_3)$ $\,\rtimes\,$ $D_4$ (2)	$(C_3^2:Q_8)$ $\,\rtimes\,$ $D_4$ (2)	$(D_6:D_6)$ $\,\rtimes\,$ $C_2^2$	$C_3^2$ $\,\rtimes\,$ $(D_4:D_4)$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6:D_6)$ . $D_4$	$C_2$ . $(D_6\wr C_2)$	$(D_6.D_6)$ . $C_2^2$ (2)	$(C_2\times C_6^2)$ . $D_4$	all 11

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 2186 subgroups in 305 conjugacy classes, 31 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_6\wr C_2$
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:	$\operatorname{Fit} \simeq$ $C_2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $D_4$
Radical:	$R \simeq$ $C_2^3.\SOPlus(4,2)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2\wr C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:D_4$
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: $C_2^3.\SOPlus(4,2)$

Order 288: $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_2^3:S_3^2$, $C_2^3.S_3^2$, $(C_2\times C_6^2).C_4$

Order 144: $C_6^2:C_2^2$ x 2, $D_6:D_6$ x 2, $D_6.D_6$ x 2, $D_6.D_6$ x 2, $C_3^2:\OD_{16}$ x 2, $C_3^2:\SD_{16}$ x 2, $C_3^2:D_8$ x 2, $C_3^2:C_4^2$, $C_6^2:C_2^2$, $D_6:D_6$, $C_6:D_{12}$, $D_6:D_6$

Order 96: $D_4:D_6$, $C_{12}:D_4$

Order 72: $C_6\wr C_2$ x 6, $C_3:D_{12}$ x 4, $D_6:S_3$ x 4, $C_6\times D_6$ x 2, $C_6^2:C_2$ x 2, $C_6:C_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_6.D_6$ x 2, $C_3^2:C_8$ x 2, $C_2\times C_6^2$, $C_6:D_6$, $S_3\times D_6$, $C_6.D_6$, $C_6.D_6$

Order 64: $D_4:D_4$

Order 48: $C_6:D_4$ x 8, $D_4:S_3$ x 4, $S_3\times D_4$ x 4, $C_6\times D_4$ x 2, $D_{12}:C_2$ x 2, $C_2\times D_{12}$, $C_{12}:C_4$

Order 36: $C_6\times S_3$ x 6, $C_3:C_{12}$ x 4, $C_6^2$ x 3, $C_3^2:C_4$ x 2, $C_6:S_3$ x 2, $S_3^2$

Order 32: $D_8:C_2$ x 2, $C_4\wr C_2$ x 2, $\OD_{16}:C_2$, $D_4:C_2^2$, $C_4:D_4$

Order 24: $C_3:D_4$ x 20, $C_6:C_4$ x 6, $C_2\times D_6$ x 6, $C_3\times D_4$ x 6, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_2^2\times C_6$ x 4, $C_2\times C_{12}$ x 2, $C_3:Q_8$ x 2

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

Order 16: $C_2\times D_4$ x 8, $D_4:C_2$ x 4, $\SD_{16}$ x 2, $D_8$ x 2, $\OD_{16}$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 14, $D_6$ x 14, $C_3:C_4$ x 8, $C_{12}$ x 4

Order 9: $C_3^2$

Order 8: $D_4$ x 16, $C_2\times C_4$ x 6, $C_2^3$ x 5, $Q_8$ x 2, $C_8$ x 2

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

Order 4: $C_2^2$ x 12, $C_4$ x 6

Order 3: $C_3$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_2^3.\SOPlus(4,2)$

Order 288: $C_2^3:S_3^2$, $C_6^2.D_4$, $C_2^3.S_3^2$, $(C_2\times C_6^2).C_4$, $C_6^2.D_4$

Order 144: $C_6^2:C_2^2$ x 2, $C_3^2:C_4^2$, $C_6^2:C_2^2$, $D_6:D_6$, $D_6:D_6$, $C_6:D_{12}$, $D_6:D_6$, $D_6.D_6$, $D_6.D_6$, $C_3^2:\OD_{16}$, $C_3^2:\SD_{16}$, $C_3^2:D_8$

Order 96: $D_4:D_6$, $C_{12}:D_4$

Order 72: $C_6\wr C_2$ x 3, $C_6\times D_6$ x 2, $C_6:C_{12}$ x 2, $C_3:D_{12}$ x 2, $D_6:S_3$ x 2, $C_2\times C_6^2$, $C_6:D_6$, $S_3\times D_6$, $C_6^2:C_2$, $C_6.D_6$, $C_3^2:Q_8$, $C_6.D_6$, $C_6.D_6$, $C_3^2:C_8$

Order 64: $D_4:D_4$

Order 48: $C_6:D_4$ x 7, $C_6\times D_4$ x 2, $D_4:S_3$ x 2, $S_3\times D_4$ x 2, $D_{12}:C_2$, $C_2\times D_{12}$, $C_{12}:C_4$

Order 36: $C_6\times S_3$ x 5, $C_6^2$ x 3, $C_3:C_{12}$ x 2, $C_6:S_3$ x 2, $C_3^2:C_4$, $S_3^2$

Order 32: $\OD_{16}:C_2$, $D_4:C_2^2$, $D_8:C_2$, $C_4:D_4$, $C_4\wr C_2$

Order 24: $C_3:D_4$ x 10, $C_6:C_4$ x 5, $C_2\times D_6$ x 5, $C_2^2\times C_6$ x 4, $C_3\times D_4$ x 3, $C_2\times C_{12}$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3:Q_8$

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

Order 16: $C_2\times D_4$ x 7, $D_4:C_2$ x 2, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_4^2$

Order 12: $C_2\times C_6$ x 13, $D_6$ x 11, $C_3:C_4$ x 4, $C_{12}$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 8, $C_2^3$ x 5, $C_2\times C_4$ x 5, $Q_8$, $C_8$

Order 6: $C_6$ x 10, $S_3$ x 4

Order 4: $C_2^2$ x 11, $C_4$ x 3

Order 3: $C_3$ x 2

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_2^3.\SOPlus(4,2)$ ($C_1$)

Order 288: $C_6^2.D_4$ ($C_2$) x 2, $C_6^2.D_4$ ($C_2$) x 2, $C_2^3:S_3^2$ ($C_2$), $C_2^3.S_3^2$ ($C_2$), $(C_2\times C_6^2).C_4$ ($C_2$)

Order 144: $D_6.D_6$ ($C_2^2$) x 2, $C_3^2:\OD_{16}$ ($C_2^2$) x 2, $C_3^2:C_4^2$ ($C_2^2$), $C_6^2:C_2^2$ ($C_2^2$), $D_6:D_6$ ($C_2^2$)

Order 72: $C_3^2:Q_8$ ($D_4$) x 2, $D_6:S_3$ ($D_4$) x 2, $C_2\times C_6^2$ ($D_4$), $C_6:D_6$ ($D_4$), $C_6.D_6$ ($C_2^3$)

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

Order 18: $C_3\times C_6$ ($C_2^2\wr C_2$)

Order 9: $C_3^2$ ($D_4:D_4$)

Order 8: $C_2^3$ ($\SOPlus(4,2)$)

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

Order 2: $C_2$ ($D_6\wr C_2$)

Order 1: $C_1$ ($C_2^3.\SOPlus(4,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_2^3.\SOPlus(4,2)$ ($C_1$)

Order 288: $C_2^3:S_3^2$ ($C_2$), $C_6^2.D_4$ ($C_2$), $C_2^3.S_3^2$ ($C_2$), $(C_2\times C_6^2).C_4$ ($C_2$), $C_6^2.D_4$ ($C_2$)

Order 144: $C_3^2:C_4^2$ ($C_2^2$), $C_6^2:C_2^2$ ($C_2^2$), $D_6:D_6$ ($C_2^2$), $D_6.D_6$ ($C_2^2$), $C_3^2:\OD_{16}$ ($C_2^2$)

Order 72: $C_2\times C_6^2$ ($D_4$), $C_6:D_6$ ($D_4$), $C_6.D_6$ ($C_2^3$), $C_3^2:Q_8$ ($D_4$), $D_6:S_3$ ($D_4$)

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

Order 18: $C_3\times C_6$ ($C_2^2\wr C_2$)

Order 9: $C_3^2$ ($D_4:D_4$)

Order 8: $C_2^3$ ($\SOPlus(4,2)$)

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

Order 2: $C_2$ ($D_6\wr C_2$)

Order 1: $C_1$ ($C_2^3.\SOPlus(4,2)$)

Series

Derived series

$C_2^3.\SOPlus(4,2)$

$\rhd$

$C_6.D_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_2^3.\SOPlus(4,2)$

$\rhd$

$C_2^3:S_3^2$

$\rhd$

$C_6^2:C_2^2$

$\rhd$

$C_6.D_6$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_2^3.\SOPlus(4,2)$

$\rhd$

$C_6.D_6$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 13 larger groups in the database.

This group is a maximal quotient of 10 larger groups in the database.

Character theory

Complex character table

See the $36 \times 36$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.