Abstract group 18432.eg: $C_2^5.\POPlus(4,3)$

• Label: 18432.eg
• Order: \( 2^{11} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2^5.\POPlus(4,3)$
Order:	\(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_{42}.D_7^2$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \)
Composition factors:	$C_2$ x 11, $C_3$ x 2
Derived length:	$4$

This group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	895	512	2688	6656	4608	3072	18432
Conjugacy classes	1	17	3	22	13	20	4	80
Divisions	1	17	3	22	13	20	4	80
Autjugacy classes	1	13	3	15	11	10	2	55

Dimension	1	2	3	4	6	9	12	18	24	36
Irr. complex chars.	8	8	8	2	8	16	10	8	4	8	80
Irr. rational chars.	8	8	8	2	8	16	10	8	4	8	80

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid b^{6}=c^{6}=d^{2}=f^{4}=h^{4}=i^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,13)(2,14)(3,10,4,9)(5,16,6,15)(7,11)(8,12)(17,24,18,23)(19,20), (1,22,2,21) \!\cdots\! \rangle$
Transitive group:	24T12458				more information
Direct product:	$C_2$ $\, \times\, $ $(C_2^4.\POPlus(4,3))$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^5$ . $\POPlus(4,3)$	$(C_2\times C_4^2.S_4)$ . $S_4$	$C_2^3$ . $(C_2^6.S_3^2)$	$C_2^2$ . $(C_2^7:S_3^2)$	all 31
Aut. group:	$\Aut((C_2\times C_4^2).S_4)$	$\Aut((C_2\times C_4^2).S_4)$	$\Aut(C_2\times C_4^2.S_4)$	$\Aut(C_2^4.\SOPlus(4,3))$	all 10

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 729660 subgroups in 14463 conjugacy classes, 41 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^4.\POPlus(4,3)$
Commutator:	$G' \simeq$ $C_2^4.A_4^2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^7:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_2^2.C_2^6$	$G/\operatorname{Fit} \simeq$ $S_3^2$
Radical:	$R \simeq$ $C_2^5.\POPlus(4,3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^6.S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.C_2^4.C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

All subgroups of index up to 128 (order at least 144) are shown, as well as all normal subgroups of any index.

Order 18432: $C_2^5.\POPlus(4,3)$

Order 9216: $C_2^4.\POPlus(4,3)$ x 4, $C_2^2.C_2^6.(C_6\times S_3)$, $C_2^5.A_4\wr C_2$, $C_2^5.\PSOPlus(4,3)$

Order 6144: $C_2^2.C_2^6.D_6.C_2$, $C_4^3.(C_2^2\times S_4)$

Order 4608: $C_4^2:A_4.A_4.C_2$ x 2, $C_2^4.\PSOPlus(4,3)$ x 2, $C_2^4.A_4\wr C_2$ x 2, $C_2^3.\POPlus(4,3)$, $C_2^5.A_4^2$

Order 3072: $C_4^2.(D_4\times S_4)$ x 8, $Q_8^2:(C_2\times S_4)$ x 4, $C_2^5.\GL(2,\mathbb{Z}/4)$ x 2, $C_4^2:A_4.C_2^4$, $C_2^2.C_2^6.D_6$, $C_4^2:A_4.C_2^3.C_2$, $C_4^2:A_4.C_2^4$, $C_4^2:A_4.C_2.C_2^3$, $C_2^2.C_2^6.C_6.C_2$, $C_2^4.C_2^4.D_6$, $C_2\times C_4^3:S_4$, $C_4^3.(C_2^2\times A_4)$

Order 2304: $C_2^2.\POPlus(4,3)$ x 4, $C_2^3.\PSOPlus(4,3)$ x 2, $C_2^4.A_4^2$, $C_2^3.A_4\wr C_2$, $C_2^3.A_4\wr C_2$

Order 2048: $C_2^3.C_2^4.C_2^4$

Order 1536: $C_4^2.\GL(2,\mathbb{Z}/4)$ x 8, $C_4^2.(C_2^2\times S_4)$ x 6, $C_4^2.(C_2^2\times S_4)$ x 4, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_4^3:S_4$ x 4, $C_4^3.S_4$ x 4, $C_4^2.(D_4\times A_4)$ x 4, $C_2^2.C_2^6.S_3$ x 2, $Q_8^2:S_4$ x 2, $C_4^2.(C_2^2\times S_4)$ x 2, $(C_2^2\times C_4^2):S_4$ x 2, $(C_2^2\times C_4^2).S_4$ x 2, $C_4^2.(C_2^2\times S_4)$ x 2, $C_4^2.(C_2^3\times A_4)$ x 2, $Q_8^2:(C_2\times A_4)$ x 2, $C_2\times C_2^3:A_4.C_2^3$, $C_2\times C_4^3:D_6$, $C_2\times Q_8^2.D_6$, $C_2\times C_2^4:A_4.C_2^2$, $C_2.C_2^6.A_4$, $C_4^2.(C_2^3\times A_4)$, $C_2\times C_4^3:A_4$

Order 1152: $C_2^2.\PSOPlus(4,3)$ x 4, $C_2^3.A_4^2$ x 2, $C_2^2.A_4\wr C_2$ x 2, $C_2^2.A_4\wr C_2$ x 2, $C_2\times S_4^2$

Order 1024: $C_4^2.D_4^2$ x 16, $C_2^2.C_2^5.C_2^3$, $C_2^2.C_2^5.C_2^3$, $C_2^3.C_2^4.C_2^3$, $C_2.C_2^6.C_2^3$, $C_4.D_4.C_2^5$, $(C_2\times C_4):D_4.C_2^4$, $C_2^2.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^2$, $C_2^3.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^2$, $C_2^2.C_2^5.C_2^3$, $C_2^3.C_2^4.C_2^3$, $C_4:Q_8.C_2^5$, $C_2^3.C_2^4.C_2^3$, $C_2^2.C_2^5.C_2^3$

Order 768: $C_2^4.(C_2\times S_4)$ x 8, $C_4^3:D_6$ x 8, $C_4^2.(C_2\times S_4)$ x 8, $C_4^2.(C_2\times S_4)$ x 7, $(C_2\times C_4^2):S_4$ x 6, $(C_2\times C_4^2).S_4$ x 4, $Q_8^2:D_6$ x 4, $C_2.C_2\wr S_4$ x 4, $C_4^2.(C_2^2\times A_4)$ x 4, $C_2\times Q_8:C_2^2.D_6$ x 3, $C_2^5:S_4$ x 3, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $C_4^2.(C_2^2\times A_4)$ x 3, $C_2^5.S_4$ x 2, $C_2\times C_4^2.S_4$ x 2, $C_2\times C_2^4.S_4$ x 2, $C_2^5:S_4$ x 2, $C_2^5.S_4$ x 2, $(C_2^2\times C_4^2):A_4$ x 2, $C_2^2.C_2\wr A_4$ x 2, $C_4^3:A_4$ x 2, $C_2\times Q_8:C_2^2.D_6$, $C_2\times Q_8:C_2^2.D_6$, $C_2\times Q_8^2.S_3$, $C_2\times C_2^4:A_4.C_2$, $C_2\times Q_8^2.S_3$, $C_4^3:D_6$, $C_4^3:D_6$, $C_2.C_2^6.C_6$, $C_2^2.C_2^6.C_3$, $C_2\times C_4^3:C_6$, $C_2\times Q_8^2.C_6$, $C_4^2.(C_2^2\times A_4)$

Order 576: $S_4^2$ x 4, $A_4^2:C_2^2$ x 2, $A_4^2:C_2^2$ x 2, $C_2^2.A_4^2$ x 2

Order 512: $C_4^3:C_2^3$ x 20, $Q_8^2:C_2^3$ x 12, $(C_4\times \OD_{16}):D_4$ x 12, $D_4^2:D_4$ x 12, $Q_8^2:C_2^3$ x 10, $(C_4\times \OD_{16}).D_4$ x 8, $Q_8^2:D_4$ x 8, $C_4^3:D_4$ x 8, $Q_8^2:D_4$ x 8, $C_4^2.(C_4\times D_4)$ x 8, $Q_8.C_2\wr C_4$ x 8, $C_4^3.D_4$ x 8, $C_4^3.D_4$ x 8, $D_4.C_2\wr C_4$ x 8, $C_4^2.C_2^4.C_2$ x 8, $C_2\times C_2^2.C_2^5.C_2$ x 4, $C_2\times C_2.D_4^2.C_2$ x 4, $C_2\times C_2^2.C_2^5.C_2$ x 3, $C_2\times (C_2^2\times C_4).C_2^4$ x 3, $C_2\times C_4.C_2^5.C_2$ x 3, $C_2\times C_2^2.C_2^5.C_2$ x 3, $D_4^2.C_2^3$ x 2, $C_2\times C_4.D_4^2$ x 2, $C_4^3:C_2^3$ x 2, $C_2\times C_4.D_4^2$ x 2, $C_2\times C_2.D_4^2.C_2$ x 2, $C_2\times C_2.D_4^2.C_2$ x 2, $C_2\times C_4:Q_8.C_2^3$ x 2, $C_2\times C_4.D_4.C_2^3$ x 2, $C_2\times C_2^2.C_2^5.C_2$ x 2, $C_2\times C_2^3.C_2^4.C_2$ x 2, $C_2\times C_4.C_2^5.C_2$ x 2, $C_2\times C_2.C_2^6.C_2$ x 2, $C_2\times C_4^2.C_2^3.C_2$, $C_2\times C_2^2.C_2^5.C_2$, $C_2\times C_2^2.C_2^5.C_2$, $C_2\times C_2^2.C_2^5.C_2$, $C_2\times C_4.D_4^2$, $C_2\times C_4^2.C_2^4$, $C_2\times C_4^2.C_2^4$, $C_2\times (C_2\times D_4:C_2).D_4$, $C_2\times (C_2\times D_4:C_2).D_4$, $C_2\times (C_2\times D_4:C_2).D_4$, $C_2\times (C_2\times D_4:C_2).D_4$, $C_2\times C_4:Q_8.C_2^3$, $C_2\times C_4.D_4.C_2^3$, $C_2\times C_4.D_4.C_2^3$, $C_2\times C_4:D_4.C_2^3$, $C_2\times C_2^3.C_2^4.C_2$, $C_2\times C_2^2.C_2^6$

Order 384: $C_2^4.S_4$ x 16, $C_2^4:S_4$ x 12, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 12, $C_2^4.S_4$ x 12, $C_2^4.S_4$ x 12, $C_2.C_2\wr A_4$ x 4, $C_4^3:C_6$ x 4, $C_4^2:D_{12}$ x 4, $C_4\wr S_3$ x 4, $C_2^4:(C_2\times A_4)$ x 4, $Q_8.(C_2\times S_4)$ x 4, $Q_8.(C_2\times S_4)$ x 4, $(C_2\times C_4^2):A_4$ x 4, $C_4^2:S_4$ x 4, $C_4^2.S_4$ x 4, $C_2^4:S_4$ x 4, $C_2^4.S_4$ x 4, $C_2^4.S_4$ x 3, $C_2^5:A_4$ x 3, $C_2^5:A_4$ x 3, $C_2^5:A_4$ x 3, $C_2^5.A_4$ x 3, $C_2^4:S_4$ x 3, $Q_8^2:C_6$ x 2, $C_4^2.(C_2\times A_4)$ x 2, $C_2^4.S_4$ x 2, $C_2^4.S_4$ x 2, $\GL(2,3):C_2^3$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^5:A_4$ x 2, $(C_2\times Q_8).S_4$ x 2, $Q_8:\GL(2,3)$ x 2, $Q_8^2:S_3$ x 2, $C_4^3:C_6$, $(C_2^3\times C_4):A_4$, $Q_8^2:C_6$

Order 288: $\PSOPlus(4,3)$ x 4, $A_4\times S_4$ x 4, $C_2\times A_4^2$ x 2, $D_6\times S_4$ x 2

Order 256: $C_4^2.(C_2\times D_4)$ x 54, $C_4.D_4^2$ x 48, $C_4.D_4^2$ x 48, $C_4^2.C_2^4$ x 42, $C_4.D_4^2$ x 40, $C_4.D_4^2$ x 40, $C_4^2.(C_2\times D_4)$ x 32, $Q_8^2:C_2^2$ x 31, $C_4^3:C_2^2$ x 28, $C_4.D_4^2$ x 26, $C_4^3:C_2^2$ x 26, $D_4^2:C_2^2$ x 21, $Q_8^2:C_2^2$ x 16, $(C_4\times C_8).D_4$ x 16, $D_4^2:C_4$ x 14, $(C_2^2\times D_4).D_4$ x 14, $C_4^2.(C_2\times D_4)$ x 12, $C_4^2.(C_2\times D_4)$ x 11, $C_4^2.C_2^4$ x 11, $C_4^2.(C_2^2\times C_4)$ x 10, $C_4^3:C_2^2$ x 10, $C_4^2:C_2^4$ x 9, $C_2\times C_4^2.D_4$ x 9, $Q_8^2:C_2^2$ x 9, $C_4^2.C_2^4$ x 8, $D_4^2:C_2^2$ x 7, $C_4.D_4^2$ x 7, $C_4^2:C_2^4$ x 7, $C_2\times C_4^2.D_4$ x 7, $C_2^5.D_4$ x 7, $C_4^2.(C_2^2\times C_4)$ x 6, $(C_4\times \OD_{16}):C_4$ x 6, $C_4^2.(C_2\times D_4)$ x 6, $Q_8^2:C_4$ x 6, $\OD_{16}.C_2^4$ x 6, $C_4^2.(C_2^2\times C_4)$ x 6, $(C_4\times \OD_{16}):C_2^2$ x 6, $Q_8^2:C_4$ x 6, $C_4^2.C_2^4$ x 6, $C_4^2.(C_2\times D_4)$ x 5, $D_8:C_2^4$ x 5, $C_2\times C_4^2.D_4$ x 4, $C_4.D_4^2$ x 4, $C_4^3.C_4$ x 4, $C_4^3:C_4$ x 4, $(C_2\times D_4).C_4^2$ x 4, $(C_2\times C_4^2).D_4$ x 4, $C_2^5:D_4$ x 4, $C_2\times \OD_{16}:D_4$ x 4, $C_4^3:C_2^2$ x 4, $C_2\times C_4^2.D_4$ x 4, $C_2\times C_4^2.D_4$ x 3, $D_4^2:C_2^2$ x 3, $D_4^2:C_2^2$ x 3, $C_4.D_4^2$ x 3, $C_4^2.C_2^4$ x 3, $C_4^2.C_2^4$ x 3, $Q_8^2:C_2^2$ x 3, $D_4^2:C_2^2$ x 3, $C_4^2.(C_2^2\times C_4)$ x 3, $(D_4\times C_2^3):C_4$ x 3, $C_2\times C_4^2.D_4$ x 3, $C_2\times \OD_{16}.D_4$ x 3, $C_2\times C_4^2.D_4$ x 3, $C_2\times \OD_{16}:D_4$ x 3, $(C_2^2\times D_8):C_4$ x 3, $C_2\times \OD_{16}:D_4$ x 3, $C_2\times \OD_{16}.D_4$ x 3, $(D_4\times C_2^3):C_4$ x 3, $C_4^2.(C_2^2\times C_4)$ x 2, $C_4^2.(C_2^2\times C_4)$ x 2, $C_2\times C_4^2.D_4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_2^3.C_2^5$ x 2, $\OD_{16}.C_2^4$ x 2, $(C_2^2\times C_4^2):C_4$ x 2, $C_2\times \OD_{16}:D_4$ x 2, $C_4^3:C_2^2$ x 2, $C_4^3:C_2^2$ x 2, $C_4^2.C_2^4$ x 2, $C_2\times C_4^2.D_4$, $C_2^4:C_2^4$, $Q_8^2:C_2^2$, $C_4^3:C_2^2$, $C_2^5.Q_8$, $C_4^2.C_2^4$, $Q_8^2:C_2^2$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$

Order 192: $C_4^2:D_6$ x 18, $C_4^2:D_6$ x 16, $\GL(2,3):C_2^2$ x 16, $D_4\times S_4$ x 16, $C_2^4.A_4$ x 12, $C_2^3:S_4$ x 9, $C_2^4:A_4$ x 8, $Q_8:S_4$ x 8, $C_2^3.S_4$ x 6, $C_2\times \GL(2,\mathbb{Z}/4)$ x 6, $C_2^4:A_4$ x 6, $\GL(2,3):C_2^2$ x 5, $C_2^3.S_4$ x 4, $C_2^3\times S_4$ x 4, $Q_8.S_4$ x 4, $C_2^2\times \GL(2,3)$ x 4, $C_2^3:S_4$ x 3, $C_2^5:C_6$ x 3, $C_2^4:A_4$ x 3, $C_2^4:A_4$ x 3, $C_2^4.A_4$ x 2, $C_4\wr C_3$ x 2, $(C_2^2\times C_4):A_4$ x 2, $\SL(2,3).C_2^3$ x 2, $\GL(2,3):C_2^2$ x 2, $\GL(2,3):C_2^2$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_4^2:A_4$ x 2, $Q_8:\SL(2,3)$

Order 144: $S_3\times S_4$ x 8, $C_6:S_4$ x 3, $A_4\times D_6$ x 2, $C_6\times S_4$ x 2, $A_4^2$ x 2

Order 128: $C_4^2:C_2^3$

Order 64: $C_4^2:C_2^2$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 128 (order at least 144) are shown, as well as all normal subgroups of any index.

Order 18432: $C_2^5.\POPlus(4,3)$

Order 9216: $C_2^2.C_2^6.(C_6\times S_3)$, $C_2^4.\POPlus(4,3)$, $C_2^5.A_4\wr C_2$, $C_2^5.\PSOPlus(4,3)$

Order 6144: $C_2^2.C_2^6.D_6.C_2$, $C_4^3.(C_2^2\times S_4)$

Order 4608: $C_4^2:A_4.A_4.C_2$, $C_2^4.\PSOPlus(4,3)$, $C_2^4.A_4\wr C_2$, $C_2^3.\POPlus(4,3)$, $C_2^5.A_4^2$

Order 3072: $C_2^5.\GL(2,\mathbb{Z}/4)$ x 2, $C_4^2.(D_4\times S_4)$ x 2, $C_4^2:A_4.C_2^4$, $C_2^2.C_2^6.D_6$, $C_4^2:A_4.C_2^3.C_2$, $C_4^2:A_4.C_2^4$, $C_4^2:A_4.C_2.C_2^3$, $C_2^2.C_2^6.C_6.C_2$, $C_2^4.C_2^4.D_6$, $C_2\times C_4^3:S_4$, $C_4^3.(C_2^2\times A_4)$, $Q_8^2:(C_2\times S_4)$

Order 2304: $C_2^3.\PSOPlus(4,3)$ x 2, $C_2^2.\POPlus(4,3)$, $C_2^4.A_4^2$, $C_2^3.A_4\wr C_2$, $C_2^3.A_4\wr C_2$

Order 2048: $C_2^3.C_2^4.C_2^4$

Order 1536: $C_4^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_4^2.(C_2^2\times S_4)$ x 2, $C_4^2.(C_2^2\times S_4)$ x 2, $(C_2^2\times C_4^2):S_4$ x 2, $(C_2^2\times C_4^2).S_4$ x 2, $C_4^2.(C_2^2\times S_4)$ x 2, $C_4^2.(C_2^2\times S_4)$ x 2, $C_4^2.(C_2^3\times A_4)$ x 2, $C_4^2.(D_4\times A_4)$ x 2, $C_2\times C_2^3:A_4.C_2^3$, $C_2^2.C_2^6.S_3$, $Q_8^2:S_4$, $C_2\times C_4^3:D_6$, $C_2\times Q_8^2.D_6$, $C_2\times C_2^4:A_4.C_2^2$, $C_4^2.\GL(2,\mathbb{Z}/4)$, $C_4^3:S_4$, $C_4^3.S_4$, $C_2.C_2^6.A_4$, $Q_8^2:(C_2\times A_4)$, $C_4^2.(C_2^3\times A_4)$, $C_2\times C_4^3:A_4$

Order 1152: $C_2^3.A_4^2$ x 2, $C_2^2.\PSOPlus(4,3)$ x 2, $C_2\times S_4^2$, $C_2^2.A_4\wr C_2$, $C_2^2.A_4\wr C_2$

Order 1024: $C_4^2.D_4^2$ x 4, $C_2^2.C_2^5.C_2^3$, $C_2^2.C_2^5.C_2^3$, $C_2^3.C_2^4.C_2^3$, $C_2.C_2^6.C_2^3$, $C_4.D_4.C_2^5$, $(C_2\times C_4):D_4.C_2^4$, $C_2^2.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^2$, $C_2^3.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^2$, $C_2^2.C_2^5.C_2^3$, $C_2^3.C_2^4.C_2^3$, $C_4:Q_8.C_2^5$, $C_2^3.C_2^4.C_2^3$, $C_2^2.C_2^5.C_2^3$

Order 768: $C_4^2.(C_2\times S_4)$ x 4, $(C_2\times C_4^2):S_4$ x 4, $C_4^2.(C_2^2\times A_4)$ x 4, $C_2\times Q_8:C_2^2.D_6$ x 3, $C_2^5:S_4$ x 3, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.(C_2\times S_4)$ x 2, $C_2^5.S_4$ x 2, $C_2\times C_4^2.S_4$ x 2, $C_2\times C_2^4.S_4$ x 2, $C_2^5:S_4$ x 2, $C_2^5.S_4$ x 2, $(C_2\times C_4^2).S_4$ x 2, $C_4^3:D_6$ x 2, $C_4^2.(C_2\times S_4)$ x 2, $(C_2^2\times C_4^2):A_4$ x 2, $C_2^2.C_2\wr A_4$ x 2, $C_4^2.(C_2^2\times A_4)$ x 2, $C_2\times Q_8:C_2^2.D_6$, $C_2\times Q_8:C_2^2.D_6$, $C_2\times Q_8^2.S_3$, $C_2\times C_2^4:A_4.C_2$, $C_2\times Q_8^2.S_3$, $C_4^3:D_6$, $C_4^3:D_6$, $Q_8^2:D_6$, $C_2.C_2\wr S_4$, $C_2.C_2^6.C_6$, $C_2^2.C_2^6.C_3$, $C_2\times C_4^3:C_6$, $C_2\times Q_8^2.C_6$, $C_4^2.(C_2^2\times A_4)$, $C_4^3:A_4$

Order 576: $A_4^2:C_2^2$ x 2, $A_4^2:C_2^2$ x 2, $C_2^2.A_4^2$ x 2, $S_4^2$

Order 512: $C_4^3:C_2^3$ x 8, $D_4^2:D_4$ x 6, $Q_8^2:C_2^3$ x 5, $C_2\times C_2^2.C_2^5.C_2$ x 4, $Q_8^2:C_2^3$ x 4, $(C_4\times \OD_{16}):D_4$ x 4, $C_2\times C_2.D_4^2.C_2$ x 4, $C_4^2.C_2^4.C_2$ x 4, $C_2\times C_2^2.C_2^5.C_2$ x 3, $C_2\times (C_2^2\times C_4).C_2^4$ x 3, $C_2\times C_4.C_2^5.C_2$ x 3, $C_2\times C_2^2.C_2^5.C_2$ x 3, $D_4^2.C_2^3$ x 2, $C_2\times C_4.D_4^2$ x 2, $C_4^3:C_2^3$ x 2, $C_2\times C_4.D_4^2$ x 2, $(C_4\times \OD_{16}).D_4$ x 2, $Q_8^2:D_4$ x 2, $C_4^3:D_4$ x 2, $Q_8^2:D_4$ x 2, $C_4^2.(C_4\times D_4)$ x 2, $Q_8.C_2\wr C_4$ x 2, $C_4^3.D_4$ x 2, $C_4^3.D_4$ x 2, $D_4.C_2\wr C_4$ x 2, $C_2\times C_2.D_4^2.C_2$ x 2, $C_2\times C_2.D_4^2.C_2$ x 2, $C_2\times C_4:Q_8.C_2^3$ x 2, $C_2\times C_4.D_4.C_2^3$ x 2, $C_2\times C_2^2.C_2^5.C_2$ x 2, $C_2\times C_2^3.C_2^4.C_2$ x 2, $C_2\times C_4.C_2^5.C_2$ x 2, $C_2\times C_2.C_2^6.C_2$ x 2, $C_2\times C_4^2.C_2^3.C_2$, $C_2\times C_2^2.C_2^5.C_2$, $C_2\times C_2^2.C_2^5.C_2$, $C_2\times C_2^2.C_2^5.C_2$, $C_2\times C_4.D_4^2$, $C_2\times C_4^2.C_2^4$, $C_2\times C_4^2.C_2^4$, $C_2\times (C_2\times D_4:C_2).D_4$, $C_2\times (C_2\times D_4:C_2).D_4$, $C_2\times (C_2\times D_4:C_2).D_4$, $C_2\times (C_2\times D_4:C_2).D_4$, $C_2\times C_4:Q_8.C_2^3$, $C_2\times C_4.D_4.C_2^3$, $C_2\times C_4.D_4.C_2^3$, $C_2\times C_4:D_4.C_2^3$, $C_2\times C_2^3.C_2^4.C_2$, $C_2\times C_2^2.C_2^6$

Order 384: $C_2^4.S_4$ x 9, $C_2^4.S_4$ x 7, $C_2^4:S_4$ x 6, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 5, $C_2^4.S_4$ x 5, $(C_2\times C_4^2):A_4$ x 4, $C_2^4.S_4$ x 4, $C_2^4.S_4$ x 3, $C_2^5:A_4$ x 3, $C_2^5:A_4$ x 3, $C_2^5:A_4$ x 3, $C_2^5.A_4$ x 3, $C_2^4:S_4$ x 3, $C_2.C_2\wr A_4$ x 2, $C_4^3:C_6$ x 2, $C_2^4.S_4$ x 2, $C_2^4:(C_2\times A_4)$ x 2, $C_2^4.S_4$ x 2, $\GL(2,3):C_2^3$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^5:A_4$ x 2, $C_4^2:S_4$ x 2, $C_4^2.S_4$ x 2, $C_2^4:S_4$ x 2, $Q_8^2:C_6$, $C_4^2.(C_2\times A_4)$, $C_4^3:C_6$, $C_4^2:D_{12}$, $C_4\wr S_3$, $(C_2^3\times C_4):A_4$, $Q_8.(C_2\times S_4)$, $Q_8.(C_2\times S_4)$, $Q_8^2:C_6$, $(C_2\times Q_8).S_4$, $Q_8:\GL(2,3)$, $Q_8^2:S_3$

Order 288: $C_2\times A_4^2$ x 2, $D_6\times S_4$ x 2, $\PSOPlus(4,3)$ x 2, $A_4\times S_4$ x 2

Order 256: $C_4^2.(C_2\times D_4)$ x 28, $C_4.D_4^2$ x 20, $C_4^2.C_2^4$ x 18, $Q_8^2:C_2^2$ x 18, $C_4.D_4^2$ x 14, $D_4^2:C_2^2$ x 14, $C_4^3:C_2^2$ x 12, $C_4.D_4^2$ x 12, $C_4.D_4^2$ x 12, $C_4^2.C_2^4$ x 11, $C_4^2:C_2^4$ x 9, $C_2\times C_4^2.D_4$ x 9, $C_4^2.(C_2\times D_4)$ x 8, $(C_2^2\times D_4).D_4$ x 8, $C_4^2.C_2^4$ x 8, $C_4.D_4^2$ x 8, $C_4^3:C_2^2$ x 8, $D_4^2:C_4$ x 7, $D_4^2:C_2^2$ x 7, $C_4.D_4^2$ x 7, $C_4^2:C_2^4$ x 7, $C_2\times C_4^2.D_4$ x 7, $C_2^5.D_4$ x 7, $\OD_{16}.C_2^4$ x 6, $C_4^2.(C_2^2\times C_4)$ x 6, $C_4^2.(C_2^2\times C_4)$ x 6, $Q_8^2:C_2^2$ x 6, $D_8:C_2^4$ x 5, $Q_8^2:C_2^2$ x 4, $(C_4\times C_8).D_4$ x 4, $C_4^2.(C_2\times D_4)$ x 4, $C_2\times C_4^2.D_4$ x 4, $C_4^2.(C_2\times D_4)$ x 4, $C_4.D_4^2$ x 4, $C_2^5:D_4$ x 4, $C_4^3:C_2^2$ x 4, $C_2\times \OD_{16}:D_4$ x 4, $(C_4\times \OD_{16}):C_2^2$ x 4, $C_2\times C_4^2.D_4$ x 4, $C_4^2.C_2^4$ x 4, $C_2\times C_4^2.D_4$ x 3, $Q_8^2:C_4$ x 3, $D_4^2:C_2^2$ x 3, $D_4^2:C_2^2$ x 3, $C_4.D_4^2$ x 3, $C_4^2.C_2^4$ x 3, $C_4^2.C_2^4$ x 3, $Q_8^2:C_2^2$ x 3, $D_4^2:C_2^2$ x 3, $C_4^2.(C_2^2\times C_4)$ x 3, $(D_4\times C_2^3):C_4$ x 3, $C_2\times C_4^2.D_4$ x 3, $C_2\times \OD_{16}.D_4$ x 3, $Q_8^2:C_4$ x 3, $C_2\times C_4^2.D_4$ x 3, $C_2\times \OD_{16}:D_4$ x 3, $(C_2^2\times D_8):C_4$ x 3, $C_2\times \OD_{16}:D_4$ x 3, $C_2\times \OD_{16}.D_4$ x 3, $(D_4\times C_2^3):C_4$ x 3, $C_4^2.(C_2^2\times C_4)$ x 2, $C_4^2.(C_2^2\times C_4)$ x 2, $(C_4\times \OD_{16}):C_4$ x 2, $C_4^2.(C_2^2\times C_4)$ x 2, $C_4^2.(C_2\times D_4)$ x 2, $C_2\times C_4^2.D_4$ x 2, $C_4^2.(C_2\times D_4)$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_2^3.C_2^5$ x 2, $\OD_{16}.C_2^4$ x 2, $(C_2^2\times C_4^2):C_4$ x 2, $C_4^3:C_2^2$ x 2, $C_2\times \OD_{16}:D_4$ x 2, $C_4^3:C_2^2$ x 2, $C_4^3:C_2^2$ x 2, $C_4^2.C_2^4$ x 2, $C_2\times C_4^2.D_4$, $C_2^4:C_2^4$, $Q_8^2:C_2^2$, $C_4^3.C_4$, $C_4^3:C_4$, $(C_2\times D_4).C_4^2$, $(C_2\times C_4^2).D_4$, $C_4^3:C_2^2$, $C_2^5.Q_8$, $C_4^2.C_2^4$, $Q_8^2:C_2^2$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$

Order 192: $C_4^2:D_6$ x 13, $C_2^4.A_4$ x 11, $C_2^4:A_4$ x 7, $C_4^2:D_6$ x 6, $C_2^3:S_4$ x 6, $C_2\times \GL(2,\mathbb{Z}/4)$ x 6, $C_2^4:A_4$ x 6, $\GL(2,3):C_2^2$ x 5, $C_2^3\times S_4$ x 4, $Q_8:S_4$ x 4, $\GL(2,3):C_2^2$ x 4, $C_2^2\times \GL(2,3)$ x 4, $D_4\times S_4$ x 4, $C_2^3.S_4$ x 3, $C_2^3:S_4$ x 3, $C_2^5:C_6$ x 3, $C_2^4:A_4$ x 3, $C_2^4:A_4$ x 3, $C_2^4.A_4$ x 2, $C_2^3.S_4$ x 2, $\SL(2,3).C_2^3$ x 2, $Q_8.S_4$ x 2, $\GL(2,3):C_2^2$ x 2, $\GL(2,3):C_2^2$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_4^2:A_4$ x 2, $C_4\wr C_3$, $(C_2^2\times C_4):A_4$, $Q_8:\SL(2,3)$

Order 144: $C_6:S_4$ x 3, $A_4\times D_6$ x 2, $C_6\times S_4$ x 2, $A_4^2$ x 2, $S_3\times S_4$ x 2

Order 128: $C_4^2:C_2^3$

Order 64: $C_4^2:C_2^2$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 18432: $C_2^5.\POPlus(4,3)$ ($C_1$)

Order 9216: $C_2^4.\POPlus(4,3)$ ($C_2$) x 4, $C_2^2.C_2^6.(C_6\times S_3)$ ($C_2$), $C_2^5.A_4\wr C_2$ ($C_2$), $C_2^5.\PSOPlus(4,3)$ ($C_2$)

Order 4608: $C_4^2:A_4.A_4.C_2$ ($C_2^2$) x 2, $C_2^4.\PSOPlus(4,3)$ ($C_2^2$) x 2, $C_2^4.A_4\wr C_2$ ($C_2^2$) x 2, $C_2^5.A_4^2$ ($C_2^2$)

Order 3072: $C_4^2:A_4.C_2^4$ ($S_3$), $C_2^2.C_2^6.D_6$ ($S_3$)

Order 2304: $C_2^4.A_4^2$ ($C_2^3$)

Order 1536: $C_2^2.C_2^6.S_3$ ($D_6$) x 2, $C_4^2.(C_2^2\times S_4)$ ($D_6$) x 2, $C_4^2.(C_2^3\times A_4)$ ($D_6$), $C_2.C_2^6.A_4$ ($D_6$)

Order 768: $C_2\times C_4^2.S_4$ ($S_4$), $C_2^2.C_2^6.C_3$ ($C_2\times D_6$), $C_4^2.(C_2^2\times A_4)$ ($C_2\times D_6$)

Order 512: $C_2\times C_2^2.C_2^6$ ($S_3^2$)

Order 384: $C_4^2.S_4$ ($C_2\times S_4$) x 2, $C_2^5:A_4$ ($C_2\times S_4$)

Order 256: $D_4^2:C_2^2$ ($S_3\times D_6$)

Order 192: $C_4^2:A_4$ ($C_2^2\times S_4$)

Order 128: $C_4^2:C_2^3$ ($S_3\times S_4$)

Order 64: $C_4^2:C_2^2$ ($D_6\times S_4$)

Order 32: $C_2^5$ ($\POPlus(4,3)$)

Order 16: $C_2^4$ ($A_4^2:C_2^3$)

Order 8: $C_2^3$ ($C_2^6.S_3^2$)

Order 4: $C_2^2$ ($C_2^7:S_3^2$)

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

Order 1: $C_1$ ($C_2^5.\POPlus(4,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 18432: $C_2^5.\POPlus(4,3)$ ($C_1$)

Order 9216: $C_2^2.C_2^6.(C_6\times S_3)$ ($C_2$), $C_2^4.\POPlus(4,3)$ ($C_2$), $C_2^5.A_4\wr C_2$ ($C_2$), $C_2^5.\PSOPlus(4,3)$ ($C_2$)

Order 4608: $C_4^2:A_4.A_4.C_2$ ($C_2^2$), $C_2^4.\PSOPlus(4,3)$ ($C_2^2$), $C_2^4.A_4\wr C_2$ ($C_2^2$), $C_2^5.A_4^2$ ($C_2^2$)

Order 3072: $C_4^2:A_4.C_2^4$ ($S_3$), $C_2^2.C_2^6.D_6$ ($S_3$)

Order 2304: $C_2^4.A_4^2$ ($C_2^3$)

Order 1536: $C_2^2.C_2^6.S_3$ ($D_6$), $C_4^2.(C_2^2\times S_4)$ ($D_6$), $C_4^2.(C_2^3\times A_4)$ ($D_6$), $C_2.C_2^6.A_4$ ($D_6$)

Order 768: $C_2\times C_4^2.S_4$ ($S_4$), $C_2^2.C_2^6.C_3$ ($C_2\times D_6$), $C_4^2.(C_2^2\times A_4)$ ($C_2\times D_6$)

Order 512: $C_2\times C_2^2.C_2^6$ ($S_3^2$)

Order 384: $C_2^5:A_4$ ($C_2\times S_4$), $C_4^2.S_4$ ($C_2\times S_4$)

Order 256: $D_4^2:C_2^2$ ($S_3\times D_6$)

Order 192: $C_4^2:A_4$ ($C_2^2\times S_4$)

Order 128: $C_4^2:C_2^3$ ($S_3\times S_4$)

Order 64: $C_4^2:C_2^2$ ($D_6\times S_4$)

Order 32: $C_2^5$ ($\POPlus(4,3)$)

Order 16: $C_2^4$ ($A_4^2:C_2^3$)

Order 8: $C_2^3$ ($C_2^6.S_3^2$)

Order 4: $C_2^2$ ($C_2^7:S_3^2$)

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

Order 1: $C_1$ ($C_2^5.\POPlus(4,3)$)

Series

Derived series

$C_2^5.\POPlus(4,3)$

$\rhd$

$C_2^4.A_4^2$

$\rhd$

$D_4^2:C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^5.\POPlus(4,3)$

$\rhd$

$C_2^2.C_2^6.(C_6\times S_3)$

$\rhd$

$C_2^5.A_4^2$

$\rhd$

$C_2^4.A_4^2$

$\rhd$

$C_4^2.(C_2^2\times A_4)$

$\rhd$

$C_2^2.C_2^6.C_3$

$\rhd$

$D_4^2:C_2^2$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^5.\POPlus(4,3)$

$\rhd$

$C_2^4.A_4^2$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $80 \times 80$ rational character table. Alternatively, you may search for characters of this group with desired properties.