Abstract group 4608.dl: $C_2^6:\SOPlus(4,2)$

• Label: 4608.dl
• Order: \( 2^{9} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $12$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2^6:\SOPlus(4,2)$
Order:	\(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$A_4^2.C_2^5.C_2^2$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 9, $C_3$ 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	399	80	1584	1392	576	576	4608
Conjugacy classes	1	17	2	18	13	2	3	56
Divisions	1	17	2	18	13	2	3	56
Autjugacy classes	1	13	2	12	9	1	3	41

Dimension	1	2	4	6	8	9	12	18	24
Irr. complex chars.	8	6	12	8	1	8	6	6	1	56
Irr. rational chars.	8	6	12	8	1	8	6	6	1	56

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$1713600$

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:	${\langle a, b, c, d, e, f, g, h, i \mid a^{2}=c^{2}=d^{6}=e^{3}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(1,2)(3,6,4,7,5,8)(9,11)(10,12), (2,4)(3,5)(7,8)(9,10)(11,12), (1,3,6,4,8,5,7,2)(9,10,11,12)\rangle$
Transitive group:	24T7380	24T7392	36T5192	36T5269	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\POPlus(4,3)$ $\,\rtimes\,$ $D_4$	$C_2^2$ $\,\rtimes\,$ $(S_4\wr C_2)$	$(S_4^2:C_2^2)$ $\,\rtimes\,$ $C_2$	$(C_2\times S_4^2)$ $\,\rtimes\,$ $C_2^2$	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(S_4^2:C_2^2)$	$C_2^5$ . $(S_3^2:C_2^2)$	$(A_4^2:C_2^3)$ . $C_2^2$	$(A_4^2:C_2^3)$ . $C_2^2$	all 7
Aut. group:	$\Aut(\GOrthPlus(4,3))$	$\Aut(C_2^5.S_3^2)$	$\Aut(A_4^2:D_4)$	$\Aut(A_4^2:Q_8)$	all 5

Elements of the group are displayed as permutations of degree 12.

Homology

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

Subgroups

There are 91732 subgroups in 2912 conjugacy classes, 32 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_4^2:C_2^2$
Commutator:	$G' \simeq$ $A_4^2:C_2^2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_4^2:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6$	$G/\operatorname{Fit} \simeq$ $\SOPlus(4,2)$
Radical:	$R \simeq$ $C_2^6:\SOPlus(4,2)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $S_3^2:C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6:D_4$
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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 4608: $C_2^6:\SOPlus(4,2)$

Order 2304: $\POPlus(4,3):C_4$ x 2, $S_4^2:C_2^2$ x 2, $\GL(2,\mathbb{Z}/4):S_4$, $C_2^4:D_6^2$, $(C_2^2\times A_4^2):C_4$

Order 1152: $A_4^2:C_2^3$ x 7, $S_4\wr C_2$ x 4, $C_2\times A_4^2:C_4$ x 2, $A_4^2:C_2^3$, $A_4^2:C_2^3$, $C_2\times S_4^2$, $A_4\times \GL(2,\mathbb{Z}/4)$, $A_4:\GL(2,\mathbb{Z}/4)$, $C_2.S_4^2$

Order 768: $C_2^5:S_4$, $A_4:D_4^2$

Order 576: $\POPlus(4,3)$ x 10, $A_4^2:C_2^2$ x 6, $A_4^2:C_2^2$ x 4, $S_4^2$ x 2, $A_4^2:C_4$ x 2, $C_2^2\times A_4^2$, $A_4^2:C_2^2$, $\GL(2,\mathbb{Z}/4):S_3$, $A_4^2:C_4$

Order 512: $C_2^6:D_4$

Order 384: $C_2^4:S_4$ x 12, $C_2^2\wr S_3$ x 2, $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_6$ x 2, $C_2^4:D_{12}$ x 2, $\GL(2,\mathbb{Z}/4):C_4$ x 2, $C_2^6:C_6$, $C_2^5:A_4$, $C_4:\GL(2,\mathbb{Z}/4)$, $C_2^2:\GL(2,\mathbb{Z}/4)$

Order 288: $A_4\wr C_2$ x 4, $\PSOPlus(4,3)$ x 3, $C_2\times A_4^2$ x 2, $A_4\times C_3:D_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $D_6\wr C_2$, $D_6:S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2\times C_6:S_4$, $D_6\times S_4$, $A_4\times S_4$

Order 256: $C_2^5.D_4$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$ x 2, $D_4^2:C_2^2$ x 2, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^6:C_4$, $C_2^5:D_4$, $C_2^2:D_4^2$, $C_2^5:D_4$, $C_2^5.D_4$

Order 192: $C_2^3:S_4$ x 16, $C_2\times \GL(2,\mathbb{Z}/4)$ x 14, $C_2^3:S_4$ x 12, $D_4\times S_4$ x 8, $C_2^4:A_4$ x 6, $C_2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^3\times S_4$ x 4, $C_2^4:C_{12}$ x 3, $C_2^5:C_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.S_4$ x 2, $C_2^4.D_6$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$, $A_4:C_4^2$, $C_2^2\wr C_3$, $A_4\times C_2^4$, $C_2^4:D_6$

Order 144: $C_6:S_4$ x 4, $S_3^2:C_2^2$ x 2, $S_3\times S_4$ x 2, $S_3^2:C_4$ x 2, $C_2^2:C_6^2$, $D_6^2$, $A_4\times D_6$, $C_6\times S_4$, $A_4^2$, $D_6:D_6$, $C_6^2:C_4$, $A_4\times C_3:C_4$, $A_4:C_{12}$

Order 128: $C_2^4:D_4$ x 17, $C_2\wr D_4$ x 8, $C_2^5:C_4$ x 6, $C_2^4.D_4$ x 3, $C_2^4.D_4$ x 3, $C_2^4:D_4$ x 3, $C_4^2:C_2^3$ x 3, $C_2.D_4^2$ x 3, $C_2^4:D_4$ x 3, $C_2^4:D_4$ x 3, $C_2^4.D_4$ x 2, $C_2^5:C_4$ x 2, $C_2\times D_4^2$ x 2, $C_2^5:C_4$ x 2, $C_2.D_4^2$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^4.D_4$, $C_2^5.C_4$, $D_4:C_2^4$, $C_2^4.D_4$, $C_2.D_4^2$

Order 96: $C_2^2\times S_4$ x 31, $\GL(2,\mathbb{Z}/4)$ x 19, $C_2^2:S_4$ x 8, $C_2^3\times A_4$ x 7, $C_2^2.S_4$ x 6, $D_4\times A_4$ x 5, $C_4:S_4$ x 5, $C_2^3:A_4$ x 4, $C_4\times S_4$ x 4, $C_2^3:A_4$ x 3, $C_2^3:C_{12}$ x 3, $D_6:D_4$ x 2, $C_2^2:D_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^3:D_6$ x 2, $D_4\times D_6$ x 2, $C_2^3.D_6$ x 2, $C_2^4:C_6$, $C_{12}:D_4$, $D_6:D_4$

Order 72: $S_3\times D_6$ x 8, $\SOPlus(4,2)$ x 4, $C_3:S_4$ x 3, $C_6\times A_4$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_6:D_6$, $C_6\times D_6$, $S_3\times A_4$, $C_3\times S_4$, $C_6\wr C_2$, $C_3:D_{12}$, $C_6.D_6$

Order 64: $C_2\wr C_2^2$ x 46, $C_2^3:D_4$ x 41, $C_2^4:C_4$ x 26, $D_4:C_2^3$ x 17, $D_4^2$ x 14, $C_2^3.D_4$ x 13, $C_2^3:D_4$ x 11, $C_2\wr C_4$ x 10, $D_4\times C_2^3$ x 10, $D_4:D_4$ x 10, $C_2^3:D_4$ x 8, $C_2^3.D_4$ x 6, $C_2^4:C_4$ x 6, $C_2^3.C_2^3$ x 4, $C_4^2:C_4$ x 4, $C_4^2:C_2^2$ x 4, $\OD_{16}:C_2^2$ x 3, $C_2^3.C_2^3$ x 3, $C_2^3.D_4$ x 3, $C_2^4:C_4$ x 3, $C_4^2:C_2^2$ x 3, $C_4^2:C_2^2$ x 3, $C_2^3.D_4$ x 2, $C_2^2:C_4^2$ x 2, $D_4:C_2^3$ x 2, $C_8:C_2^3$ x 2, $D_4:D_4$ x 2, $C_2^2:\OD_{16}$, $C_2^3.Q_8$, $C_2^6$, $Q_8:D_4$, $D_4:D_4$

Order 48: $C_2\times S_4$ x 55, $C_2^2\times A_4$ x 20, $C_6:D_4$ x 14, $S_3\times D_4$ x 8, $A_4:C_4$ x 5, $C_2^2\times D_6$ x 4, $D_6:C_4$ x 4, $C_6\times D_4$ x 3, $C_2\times D_{12}$ x 3, $C_4\times A_4$ x 3, $C_2^2:C_{12}$ x 3, $C_6.C_2^3$ x 2, $C_4\times D_6$ x 2, $C_6.D_4$ x 2, $C_2^3\times C_6$, $C_2^2:A_4$, $C_6.D_4$, $C_{12}:C_4$

Order 36: $S_3^2$ x 12, $C_6\times S_3$ x 7, $C_6:S_3$ x 4, $C_3^2:C_4$ x 2, $C_3:C_{12}$, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 131, $C_2^2\wr C_2$ x 93, $D_4:C_2^2$ x 76, $C_2^3:C_4$ x 56, $C_2^3:C_4$ x 40, $D_4:C_2^2$ x 38, $C_4:D_4$ x 28, $C_2^5$ x 13, $C_4\times D_4$ x 12, $C_2^3\times C_4$ x 10, $C_4:D_4$ x 10, $C_2.C_4^2$ x 9, $D_8:C_2$ x 8, $C_4\wr C_2$ x 8, $\OD_{16}:C_2$ x 6, $C_2^2.D_4$ x 5, $C_2\times C_4^2$ x 3, $D_4:C_4$ x 2, $C_2^2:C_8$ x 2, $C_2\times \SD_{16}$ x 2, $C_2\times D_8$ x 2, $C_2\times \OD_{16}$ x 2, $Q_8:C_4$ x 2, $C_2^2\times Q_8$

Order 24: $C_2\times D_6$ x 30, $C_3:D_4$ x 19, $S_4$ x 17, $C_2\times A_4$ x 16, $C_2^2\times C_6$ x 7, $C_6:C_4$ x 6, $D_{12}$ x 5, $C_3\times D_4$ x 5, $C_4\times S_3$ x 4, $C_2\times C_{12}$ x 3

Order 18: $C_3\times S_3$ x 5, $C_3:S_3$ x 3, $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$ x 371, $D_4:C_2$ x 100, $C_2^2:C_4$ x 78, $C_2^4$ x 77, $C_2^2\times C_4$ x 65, $C_4:C_4$ x 8, $C_2\times Q_8$ x 7, $C_4^2$ x 5, $\SD_{16}$ x 4, $D_8$ x 4, $\OD_{16}$ x 4, $C_2\times C_8$ x 2

Order 12: $D_6$ x 49, $C_2\times C_6$ x 19, $C_3:C_4$ x 5, $A_4$ x 3, $C_{12}$ x 3

Order 9: $C_3^2$

Order 8: $C_2^3$ x 157, $D_4$ x 144, $C_2\times C_4$ x 95, $Q_8$ x 10, $C_8$ x 2

Order 6: $C_6$ x 13, $S_3$ x 13

Order 4: $C_2^2$ x 96, $C_4$ x 18

Order 3: $C_3$ x 2

Order 2: $C_2$ x 17

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4608: $C_2^6:\SOPlus(4,2)$

Order 2304: $\GL(2,\mathbb{Z}/4):S_4$, $C_2^4:D_6^2$, $\POPlus(4,3):C_4$, $(C_2^2\times A_4^2):C_4$, $S_4^2:C_2^2$

Order 1152: $A_4^2:C_2^3$ x 3, $A_4^2:C_2^3$, $A_4^2:C_2^3$, $C_2\times S_4^2$, $C_2\times A_4^2:C_4$, $S_4\wr C_2$, $A_4\times \GL(2,\mathbb{Z}/4)$, $A_4:\GL(2,\mathbb{Z}/4)$, $C_2.S_4^2$

Order 768: $C_2^5:S_4$, $A_4:D_4^2$

Order 576: $A_4^2:C_2^2$ x 3, $\POPlus(4,3)$ x 3, $A_4^2:C_2^2$ x 2, $C_2^2\times A_4^2$, $A_4^2:C_2^2$, $S_4^2$, $A_4^2:C_4$, $\GL(2,\mathbb{Z}/4):S_3$, $A_4^2:C_4$

Order 512: $C_2^6:D_4$

Order 384: $C_2^4:S_4$ x 3, $C_2^2\wr S_3$ x 2, $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_6$ x 2, $C_2^4:D_{12}$ x 2, $\GL(2,\mathbb{Z}/4):C_4$ x 2, $C_2^6:C_6$, $C_2^5:A_4$, $C_4:\GL(2,\mathbb{Z}/4)$, $C_2^2:\GL(2,\mathbb{Z}/4)$

Order 288: $C_2\times A_4^2$ x 2, $\PSOPlus(4,3)$ x 2, $A_4\times C_3:D_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $D_6\wr C_2$, $D_6:S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2\times C_6:S_4$, $D_6\times S_4$, $A_4\wr C_2$, $A_4\times S_4$

Order 256: $C_2^5.D_4$, $C_2^5:D_4$, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^6:C_4$, $C_2^5:D_4$, $C_2^2:D_4^2$, $D_4^2:C_2^2$, $C_2^5:D_4$, $C_2^5.D_4$

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

Order 144: $C_6:S_4$ x 3, $C_2^2:C_6^2$, $D_6^2$, $A_4\times D_6$, $C_6\times S_4$, $S_3^2:C_2^2$, $A_4^2$, $S_3\times S_4$, $D_6:D_6$, $C_6^2:C_4$, $A_4\times C_3:C_4$, $A_4:C_{12}$, $S_3^2:C_4$

Order 128: $C_2^4:D_4$ x 8, $C_2^5:C_4$ x 3, $C_2^4:D_4$ x 3, $C_2.D_4^2$ x 3, $C_2^4:D_4$ x 3, $C_2^4:D_4$ x 3, $C_2\wr D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^5:C_4$ x 2, $C_2\times D_4^2$ x 2, $C_4^2:C_2^3$ x 2, $C_2^5:C_4$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^5.C_4$, $D_4:C_2^4$, $C_2^4.D_4$, $C_2.D_4^2$, $C_2^4.D_4$, $C_2^3.C_4^2$

Order 96: $C_2^2\times S_4$ x 20, $\GL(2,\mathbb{Z}/4)$ x 14, $C_2^3\times A_4$ x 7, $D_4\times A_4$ x 5, $C_2^2.S_4$ x 5, $C_4:S_4$ x 4, $C_2^2:S_4$ x 3, $C_2^3:C_{12}$ x 3, $C_4\times S_4$ x 3, $D_6:D_4$ x 2, $C_2^2:D_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^3:A_4$ x 2, $C_2^3:D_6$ x 2, $D_4\times D_6$ x 2, $C_2^3.D_6$ x 2, $C_2^3:A_4$, $C_2^4:C_6$, $C_{12}:D_4$, $D_6:D_4$

Order 72: $S_3\times D_6$ x 4, $C_6\times A_4$ x 2, $C_3:S_4$ x 2, $C_6:D_6$, $C_6\times D_6$, $C_2\times C_3^2:C_4$, $S_3\times A_4$, $C_3\times S_4$, $\SOPlus(4,2)$, $C_6\wr C_2$, $C_3:D_{12}$, $C_6.D_6$

Order 64: $C_2^3:D_4$ x 29, $C_2^3.D_4$ x 12, $C_2\wr C_2^2$ x 12, $D_4^2$ x 11, $C_2^4:C_4$ x 10, $C_2^3:D_4$ x 10, $D_4\times C_2^3$ x 10, $D_4:C_2^3$ x 8, $C_2^3:D_4$ x 8, $C_2^3.D_4$ x 6, $C_2^4:C_4$ x 6, $C_2^3.C_2^3$ x 4, $C_2\wr C_4$ x 4, $C_2^3.C_2^3$ x 3, $C_2^3.D_4$ x 3, $C_2^4:C_4$ x 3, $C_4^2:C_2^2$ x 3, $C_4^2:C_2^2$ x 3, $D_4:D_4$ x 3, $\OD_{16}:C_2^2$ x 2, $C_2^2:C_4^2$ x 2, $D_4:C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.D_4$, $C_2^2:\OD_{16}$, $C_2^3.Q_8$, $C_4^2:C_4$, $C_2^6$, $C_8:C_2^3$, $D_4:D_4$, $Q_8:D_4$, $D_4:D_4$

Order 48: $C_2\times S_4$ x 27, $C_2^2\times A_4$ x 16, $C_6:D_4$ x 12, $S_3\times D_4$ x 6, $C_2^2\times D_6$ x 4, $A_4:C_4$ x 4, $D_6:C_4$ x 4, $C_6\times D_4$ x 3, $C_2\times D_{12}$ x 3, $C_4\times A_4$ x 3, $C_2^2:C_{12}$ x 3, $C_6.C_2^3$ x 2, $C_4\times D_6$ x 2, $C_6.D_4$ x 2, $C_2^3\times C_6$, $C_2^2:A_4$, $C_6.D_4$, $C_{12}:C_4$

Order 36: $S_3^2$ x 4, $C_6:S_3$ x 3, $C_6\times S_3$ x 3, $C_3^2:C_4$, $C_3:C_{12}$, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 87, $C_2^2\wr C_2$ x 56, $C_2^3:C_4$ x 45, $C_4:D_4$ x 24, $D_4:C_2^2$ x 18, $C_2^5$ x 13, $D_4:C_2^2$ x 12, $C_2^3:C_4$ x 11, $C_2^3\times C_4$ x 10, $C_4\times D_4$ x 10, $C_2.C_4^2$ x 8, $C_4:D_4$ x 7, $C_2^2.D_4$ x 5, $\OD_{16}:C_2$ x 3, $C_2\times C_4^2$ x 3, $C_2^2:C_8$ x 2, $D_8:C_2$ x 2, $C_4\wr C_2$ x 2, $D_4:C_4$, $C_2^2\times Q_8$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_2\times \OD_{16}$, $Q_8:C_4$

Order 24: $C_2\times D_6$ x 19, $C_3:D_4$ x 14, $C_2\times A_4$ x 11, $S_4$ x 9, $C_2^2\times C_6$ x 7, $C_6:C_4$ x 5, $C_3\times D_4$ x 5, $D_{12}$ x 4, $C_2\times C_{12}$ x 3, $C_4\times S_3$ x 3

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

Order 16: $C_2\times D_4$ x 182, $C_2^4$ x 65, $C_2^2:C_4$ x 55, $C_2^2\times C_4$ x 43, $D_4:C_2$ x 19, $C_4:C_4$ x 7, $C_4^2$ x 4, $C_2\times Q_8$ x 3, $\OD_{16}$ x 2, $\SD_{16}$, $D_8$, $C_2\times C_8$

Order 12: $D_6$ x 24, $C_2\times C_6$ x 15, $C_3:C_4$ x 4, $A_4$ x 3, $C_{12}$ x 3

Order 9: $C_3^2$

Order 8: $C_2^3$ x 115, $D_4$ x 68, $C_2\times C_4$ x 53, $Q_8$ x 3, $C_8$

Order 6: $C_6$ x 9, $S_3$ x 7

Order 4: $C_2^2$ x 69, $C_4$ x 12

Order 3: $C_3$ x 2

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4608: $C_2^6:\SOPlus(4,2)$ ($C_1$)

Order 2304: $\POPlus(4,3):C_4$ ($C_2$) x 2, $S_4^2:C_2^2$ ($C_2$) x 2, $\GL(2,\mathbb{Z}/4):S_4$ ($C_2$), $C_2^4:D_6^2$ ($C_2$), $(C_2^2\times A_4^2):C_4$ ($C_2$)

Order 1152: $A_4^2:C_2^3$ ($C_2^2$) x 2, $C_2\times A_4^2:C_4$ ($C_2^2$) x 2, $A_4^2:C_2^3$ ($C_2^2$), $C_2\times S_4^2$ ($C_2^2$), $C_2.S_4^2$ ($C_2^2$)

Order 576: $\POPlus(4,3)$ ($D_4$) x 4, $C_2^2\times A_4^2$ ($D_4$), $A_4^2:C_2^2$ ($C_2^3$), $A_4^2:C_2^2$ ($D_4$)

Order 288: $\PSOPlus(4,3)$ ($C_2\times D_4$) x 2, $C_2\times A_4^2$ ($C_2\times D_4$)

Order 144: $A_4^2$ ($C_2^2\wr C_2$)

Order 64: $C_2^6$ ($\SOPlus(4,2)$)

Order 32: $C_2^5$ ($S_3^2:C_2^2$)

Order 16: $C_2^4$ ($D_6\wr C_2$)

Order 4: $C_2^2$ ($S_4\wr C_2$)

Order 2: $C_2$ ($S_4^2:C_2^2$)

Order 1: $C_1$ ($C_2^6:\SOPlus(4,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4608: $C_2^6:\SOPlus(4,2)$ ($C_1$)

Order 2304: $\GL(2,\mathbb{Z}/4):S_4$ ($C_2$), $C_2^4:D_6^2$ ($C_2$), $\POPlus(4,3):C_4$ ($C_2$), $(C_2^2\times A_4^2):C_4$ ($C_2$), $S_4^2:C_2^2$ ($C_2$)

Order 1152: $A_4^2:C_2^3$ ($C_2^2$), $A_4^2:C_2^3$ ($C_2^2$), $C_2\times S_4^2$ ($C_2^2$), $C_2\times A_4^2:C_4$ ($C_2^2$), $C_2.S_4^2$ ($C_2^2$)

Order 576: $C_2^2\times A_4^2$ ($D_4$), $A_4^2:C_2^2$ ($C_2^3$), $A_4^2:C_2^2$ ($D_4$), $\POPlus(4,3)$ ($D_4$)

Order 288: $C_2\times A_4^2$ ($C_2\times D_4$), $\PSOPlus(4,3)$ ($C_2\times D_4$)

Order 144: $A_4^2$ ($C_2^2\wr C_2$)

Order 64: $C_2^6$ ($\SOPlus(4,2)$)

Order 32: $C_2^5$ ($S_3^2:C_2^2$)

Order 16: $C_2^4$ ($D_6\wr C_2$)

Order 4: $C_2^2$ ($S_4\wr C_2$)

Order 2: $C_2$ ($S_4^2:C_2^2$)

Order 1: $C_1$ ($C_2^6:\SOPlus(4,2)$)

Series

Derived series

$C_2^6:\SOPlus(4,2)$

$\rhd$

$A_4^2:C_2^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2^6:\SOPlus(4,2)$

$\rhd$

$\GL(2,\mathbb{Z}/4):S_4$

$\rhd$

$A_4^2:C_2^3$

$\rhd$

$A_4^2:C_2^2$

$\rhd$

$C_2\times A_4^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$C_2^6:\SOPlus(4,2)$

$\rhd$

$A_4^2:C_2^2$

$\rhd$

$A_4^2$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 14 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 $56 \times 56$ rational character table. Alternatively, you may search for characters of this group with desired properties.