Abstract group 6144.cs: $C_2^6.(C_4\times S_4)$

• Label: 6144.cs
• Order: \( 2^{11} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $2$

Group information

Description:	$C_2^6.(C_4\times S_4)$
Order:	\(6144\)\(\medspace = 2^{11} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^7.(D_4\times S_4)$, of order \(24576\)\(\medspace = 2^{13} \cdot 3 \)
Composition factors:	$C_2$ x 11, $C_3$
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	367	128	2576	896	1152	1024	6144
Conjugacy classes	1	19	1	30	5	3	4	63
Divisions	1	19	1	22	5	2	2	52
Autjugacy classes	1	16	1	15	4	2	1	40

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	8	4	8	8	8	8	12	2	5	63
Irr. rational chars.	4	4	4	5	8	8	9	3	7	52

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$2$
Inequivalent generating pairs:	$288$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid b^{2}=c^{3}=d^{2}=e^{2}=f^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,12,13,6,4,9,16,8)(2,11,14,5,3,10,15,7), (1,10,5,14,2,9,6,13)(3,11,7,16,4,12,8,15)\rangle$
Transitive group:	16T1661	16T1667			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^7.S_4)$ $\,\rtimes\,$ $C_2$	$(C_2^6:S_4)$ $\,\rtimes\,$ $C_4$	$(C_2^6.S_4)$ $\,\rtimes\,$ $C_4$	$(C_2^6:C_4)$ $\,\rtimes\,$ $S_4$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^7:S_4)$ . $C_2$	$C_2^7$ . $(C_2\times S_4)$	$C_2^6$ . $(C_4\times S_4)$	$C_2^4$ . $(C_2\wr S_4)$	all 12

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 181542 subgroups in 5724 conjugacy classes, 22 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^5.(C_4\times S_4)$
Commutator:	$G' \simeq$ $C_2^6:A_4$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^7$	$G/\Phi \simeq$ $C_2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^7.C_2^3$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^6.(C_4\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2^5.(C_4\times S_4)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2^2\times D_4^2).D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 6144: $C_2^6.(C_4\times S_4)$

Order 3072: $C_2^7:S_4$, $C_2^7.S_4$, $C_2^6.(C_4\times A_4)$

Order 2048: $(C_2^2\times D_4^2).D_4$

Order 1536: $C_2^6.S_4$, $C_2^6:S_4$, $C_2^7.D_6$, $C_2^7:A_4$

Order 1024: $C_2^6.C_2^4$, $C_2^7.D_4$, $C_2^6.C_4^2$, $C_2^7.D_4$, $C_2^7.C_2^3$, $C_2^7.D_4$, $C_2^7:D_4$

Order 768: $C_2^6.D_6$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:S_4$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^5:S_4$, $C_2^4.\GL(2,3)$, $C_2^6:A_4$, $C_2^6:C_{12}$

Order 512: $C_2^6.D_4$ x 3, $C_2^6.D_4$ x 2, $C_2^6.D_4$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$ x 2, $C_2^7:C_2^2$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $C_2^6.D_4$, $C_2^6:D_4$, $C_2^6.D_4$, $C_2^6.D_4$, $C_2^6.D_4$, $D_4^2:D_4$, $C_2^7.C_4$, $C_2^7:C_4$, $C_2^6.C_2^3$, $C_2^6:D_4$, $C_2^6.D_4$

Order 384: $C_2^2\wr S_3$ x 10, $C_2\wr S_4$ x 4, $C_2^5.D_6$ x 4, $C_2^5:C_{12}$ x 2, $C_2^4.S_4$ x 2, $C_2^4:S_4$ x 2, $C_2^4:\SL(2,3)$, $C_2^5:A_4$, $C_2^5:A_4$, $C_2^4.S_4$, $C_2^3:\GL(2,3)$, $C_2^5.D_6$

Order 256: $C_2^2.D_4^2$ x 8, $C_2^5.D_4$ x 5, $C_2^6:C_4$ x 4, $C_2^5.D_4$ x 4, $C_2^4.C_4^2$ x 4, $C_2^5:D_4$ x 4, $C_2^5.D_4$ x 4, $C_2^4.C_4^2$ x 4, $C_2^6:C_4$ x 3, $C_2^6:C_4$ x 3, $C_2^6:C_2^2$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $D_4^2:C_4$ x 2, $C_2^4:\OD_{16}$ x 2, $C_2^4.Q_{16}$ x 2, $C_2^4.D_8$ x 2, $C_2^4.C_4^2$ x 2, $C_2^3.C_2^5$ x 2, $D_4^2:C_2^2$ x 2, $C_2^4.C_2^4$ x 2, $C_2^6.C_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5:D_4$, $C_2^5:D_4$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^5.D_4$, $C_4^2:C_2^4$, $C_2^2\times D_4^2$, $C_2^5:D_4$, $C_2^5.D_4$, $C_2^5:D_4$, $C_2^5.D_4$, $C_2^5:D_4$, $C_2^5.D_4$, $C_4.D_4^2$, $C_2^6:C_4$, $C_2^4.C_2^4$, $C_2^5.D_4$, $C_2^4.C_2^4$, $C_2^5.D_4$, $C_2^5.D_4$

Order 192: $C_2^3:S_4$ x 18, $C_2^2\wr C_3$ x 5, $C_2^3:S_4$ x 4, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\wr A_4$ x 2, $C_2^4:C_{12}$ x 2, $C_2^3\times S_4$ x 2, $C_2^4:A_4$ x 2, $C_2^3.S_4$ x 2, $Q_8:S_4$ x 2, $C_2^4.D_6$ x 2, $C_2^3:\SL(2,3)$ x 2, $C_2^4:C_{12}$, $C_2^4:A_4$

Order 128: $C_2^4:D_4$ x 30, $C_2^4.D_4$ x 23, $C_2^4.D_4$ x 21, $C_2^5:C_4$ x 21, $C_2^5:C_4$ x 18, $C_2^4.D_4$ x 12, $C_2^5:C_4$ x 12, $C_2.D_4^2$ x 12, $C_2\times D_4^2$ x 11, $C_2\wr D_4$ x 8, $C_2^4.C_2^3$ x 8, $C_2^3.C_4^2$ x 8, $C_2^4:D_4$ x 8, $C_2^5:C_4$ x 7, $C_2^4:D_4$ x 6, $C_2^4:D_4$ x 6, $C_2^4.D_4$ x 5, $C_2^4.D_4$ x 4, $C_2^4.C_2^3$ x 4, $C_2^3:C_4^2$ x 4, $C_2^3.C_4^2$ x 4, $C_2^4:D_4$ x 4, $C_2^3\wr C_2$ x 4, $C_2^4.D_4$ x 4, $C_2^4:C_8$ x 3, $D_4\times C_2^4$ x 3, $C_2^3:\OD_{16}$ x 3, $C_2^4.D_4$ x 2, $C_2^4:Q_8$ x 2, $C_2^3.D_8$ x 2, $C_2^5.C_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_4^2:C_2^3$ x 2, $D_4^2:C_2$ x 2, $C_2^4:D_4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^5:C_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3:C_4^2$ x 2, $C_2^3.C_4^2$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_4^2:D_4$, $C_4^2:D_4$, $C_2^4.D_4$, $C_2^3:\SD_{16}$, $C_2^3:D_8$, $C_2^7$, $D_4^2:C_2$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_2^4.D_4$, $C_2^4.D_4$

Order 96: $C_2^2\times S_4$ x 25, $C_2^2:S_4$ x 6, $C_2^3:A_4$ x 5, $C_2^2.S_4$ x 4, $C_4\times S_4$ x 4, $C_2^3\times A_4$ x 3, $Q_8:A_4$ x 2, $C_2^3:C_{12}$ x 2, $D_6.D_4$, $C_2\times \GL(2,3)$

Order 64: $C_2^3:D_4$ x 189, $C_2^4:C_4$ x 76, $D_4\times C_2^3$ x 75, $C_2^4:C_4$ x 73, $C_2^4:C_4$ x 57, $D_4^2$ x 39, $C_2^3.D_4$ x 34, $C_2^2.C_4^2$ x 33, $C_2^3:D_4$ x 20, $C_2^3:D_4$ x 20, $C_4^2:C_2^2$ x 18, $C_2^6$ x 15, $C_2\wr C_4$ x 14, $C_4^2:C_2^2$ x 14, $C_2\wr C_2^2$ x 12, $C_2^3.C_2^3$ x 8, $C_2^2.C_4^2$ x 8, $C_2^3.D_4$ x 6, $C_2^2:C_4^2$ x 6, $C_2^3:C_8$ x 6, $D_4:D_4$ x 6, $C_4^2:C_2^2$ x 6, $C_2^3:D_4$ x 5, $\OD_{16}:C_2^2$ x 4, $C_4^2:C_4$ x 4, $C_4^2:C_2^2$ x 4, $D_4:D_4$ x 4, $C_2^4\times C_4$ x 3, $C_4^2:C_2^2$ x 3, $C_2^2.Q_{16}$ x 2, $C_2^2:\OD_{16}$ x 2, $C_2^2.D_8$ x 2, $C_2^3.D_4$ x 2, $D_4:C_2^3$ x 2, $C_2^3:Q_8$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^2:\SD_{16}$ x 2, $D_4:D_4$ x 2, $C_2^3.D_4$, $C_8:C_2^3$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_4$, $C_2^3.D_4$, $C_2^2\times C_4^2$, $C_4^2:C_2^2$

Order 48: $C_2\times S_4$ x 52, $C_2^2\times A_4$ x 18, $A_4:C_4$ x 4, $C_4\times D_6$ x 2, $C_2\times \SL(2,3)$ x 2, $C_4\times A_4$ x 2, $\GL(2,3)$ x 2, $D_6:C_4$ x 2, $C_2^2\times D_6$, $C_2^2:A_4$, $C_2^2:C_{12}$, $C_6.D_4$

Order 32: $C_2^2\times D_4$ x 448, $C_2^3:C_4$ x 323, $C_2^2\wr C_2$ x 243, $C_2^5$ x 138, $C_4\times D_4$ x 54, $C_2^3\times C_4$ x 51, $C_2^3:C_4$ x 50, $C_4:D_4$ x 42, $C_2.C_4^2$ x 20, $C_4:D_4$ x 11, $C_4^2:C_2$ x 10, $C_2^2.D_4$ x 9, $C_2\times C_4^2$ x 7, $\OD_{16}:C_2$ x 6, $C_2^2.D_4$ x 6, $C_2^2:C_8$ x 4, $D_4:C_2^2$ x 4, $C_2^2:Q_8$ x 4, $D_8:C_2$ x 3, $C_4^2:C_2$ x 3, $D_4:C_4$ x 2, $D_4:C_2^2$ x 2, $C_2\times \OD_{16}$ x 2, $C_4^2:C_2$ x 2, $C_4\wr C_2$ x 2, $C_2\times \SD_{16}$, $C_2\times D_8$

Order 24: $C_2\times A_4$ x 20, $S_4$ x 18, $C_2\times D_6$ x 10, $C_4\times S_3$ x 4, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $\SL(2,3)$ x 2, $C_2^2\times C_6$

Order 16: $C_2\times D_4$ x 644, $C_2^4$ x 518, $C_2^2:C_4$ x 218, $C_2^2\times C_4$ x 180, $C_4:C_4$ x 16, $C_4^2$ x 12, $\OD_{16}$ x 4, $D_4:C_2$ x 4, $\SD_{16}$ x 2, $D_8$ x 2, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2

Order 12: $D_6$ x 18, $C_2\times C_6$ x 5, $A_4$ x 4, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 8: $C_2^3$ x 580, $D_4$ x 166, $C_2\times C_4$ x 145, $Q_8$ x 2, $C_8$ x 2

Order 6: $S_3$ x 6, $C_6$ x 5

Order 4: $C_2^2$ x 183, $C_4$ x 22

Order 3: $C_3$

Order 2: $C_2$ x 19

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6144: $C_2^6.(C_4\times S_4)$

Order 3072: $C_2^7:S_4$, $C_2^7.S_4$, $C_2^6.(C_4\times A_4)$

Order 2048: $(C_2^2\times D_4^2).D_4$

Order 1536: $C_2^6.S_4$, $C_2^6:S_4$, $C_2^7.D_6$, $C_2^7:A_4$

Order 1024: $C_2^6.C_2^4$, $C_2^7.D_4$, $C_2^6.C_4^2$, $C_2^7.D_4$, $C_2^7.C_2^3$, $C_2^7.D_4$, $C_2^7:D_4$

Order 768: $C_2^5:S_4$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^5:S_4$, $C_2^6.D_6$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^4.\GL(2,3)$, $C_2^6:A_4$, $C_2^6:C_{12}$

Order 512: $C_2^6.D_4$ x 3, $C_2^6.D_4$ x 2, $C_2^6.D_4$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$ x 2, $C_2^7:C_2^2$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $C_2^6.D_4$, $C_2^6:D_4$, $C_2^6.D_4$, $C_2^6.D_4$, $C_2^6.D_4$, $D_4^2:D_4$, $C_2^7.C_4$, $C_2^7:C_4$, $C_2^6.C_2^3$, $C_2^6:D_4$, $C_2^6.D_4$

Order 384: $C_2^2\wr S_3$ x 7, $C_2\wr S_4$ x 4, $C_2^4:S_4$ x 2, $C_2^4:\SL(2,3)$, $C_2^5:A_4$, $C_2^5:C_{12}$, $C_2^5.D_6$, $C_2^5:A_4$, $C_2^4.S_4$, $C_2^4.S_4$, $C_2^3:\GL(2,3)$, $C_2^5.D_6$

Order 256: $C_2^5.D_4$ x 5, $C_2^6:C_4$ x 4, $C_2^4.C_4^2$ x 4, $C_2^5:D_4$ x 4, $C_2^2.D_4^2$ x 4, $C_2^4.C_4^2$ x 4, $C_2^6:C_4$ x 3, $C_2^6:C_4$ x 3, $C_2^6:C_2^2$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $D_4^2:C_4$ x 2, $C_2^4:\OD_{16}$ x 2, $C_2^4.Q_{16}$ x 2, $C_2^5.D_4$ x 2, $C_2^4.D_8$ x 2, $C_2^4.C_4^2$ x 2, $C_2^3.C_2^5$ x 2, $D_4^2:C_2^2$ x 2, $C_2^4.C_2^4$ x 2, $C_2^6.C_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5:D_4$, $C_2^5:D_4$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^5.D_4$, $C_4^2:C_2^4$, $C_2^2\times D_4^2$, $C_2^5:D_4$, $C_2^5.D_4$, $C_2^5:D_4$, $C_2^5.D_4$, $C_2^5:D_4$, $C_2^5.D_4$, $C_4.D_4^2$, $C_2^6:C_4$, $C_2^4.C_2^4$, $C_2^5.D_4$, $C_2^4.C_2^4$, $C_2^5.D_4$, $C_2^5.D_4$

Order 192: $C_2^3:S_4$ x 10, $C_2^2\wr C_3$ x 4, $C_2^3:S_4$ x 3, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\wr A_4$ x 2, $C_2^3\times S_4$ x 2, $C_2^4:A_4$ x 2, $Q_8:S_4$ x 2, $C_2^3:\SL(2,3)$ x 2, $C_2^4:C_{12}$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2^4:C_{12}$, $C_2^3.S_4$, $C_2^4.D_6$, $C_2^4:A_4$

Order 128: $C_2^4:D_4$ x 24, $C_2^5:C_4$ x 17, $C_2^5:C_4$ x 16, $C_2^4.D_4$ x 13, $C_2^4.D_4$ x 12, $C_2\times D_4^2$ x 11, $C_2^4.D_4$ x 9, $C_2\wr D_4$ x 8, $C_2.D_4^2$ x 8, $C_2^5:C_4$ x 6, $C_2^5:C_4$ x 6, $C_2^4:D_4$ x 6, $C_2^4:D_4$ x 5, $C_2^4.D_4$ x 5, $C_2^4.C_2^3$ x 4, $C_2^4.D_4$ x 4, $C_2^3.C_4^2$ x 4, $C_2^3.C_4^2$ x 4, $C_2^4:D_4$ x 4, $C_2^3\wr C_2$ x 4, $C_2^4.D_4$ x 4, $C_2^4:C_8$ x 3, $D_4\times C_2^4$ x 3, $C_2^3:\OD_{16}$ x 3, $C_2^4.D_4$ x 2, $C_2^4:Q_8$ x 2, $C_2^4.C_2^3$ x 2, $C_2^5.C_4$ x 2, $C_2^3:C_4^2$ x 2, $C_4^2:C_2^3$ x 2, $D_4^2:C_2$ x 2, $C_2^4:D_4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^5:C_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3:C_4^2$ x 2, $C_2^3.C_4^2$ x 2, $C_2^4:D_4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.D_8$, $C_4^2:D_4$, $C_4^2:D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^3:\SD_{16}$, $C_2^3:D_8$, $C_2^4.D_4$, $C_2^7$, $D_4^2:C_2$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_2^4.D_4$, $C_2^4.D_4$

Order 96: $C_2^2\times S_4$ x 16, $C_2^3:A_4$ x 4, $C_2^2:S_4$ x 4, $C_2^3\times A_4$ x 3, $Q_8:A_4$ x 2, $C_2^2.S_4$ x 2, $D_6.D_4$, $C_2^3:C_{12}$, $C_2\times \GL(2,3)$, $C_4\times S_4$

Order 64: $C_2^3:D_4$ x 106, $D_4\times C_2^3$ x 59, $C_2^4:C_4$ x 59, $C_2^4:C_4$ x 54, $D_4^2$ x 38, $C_2^4:C_4$ x 29, $C_2^3:D_4$ x 20, $C_2^2.C_4^2$ x 15, $C_2^3.D_4$ x 14, $C_4^2:C_2^2$ x 14, $C_2^6$ x 13, $C_2\wr C_4$ x 11, $C_2^3:D_4$ x 10, $C_2\wr C_2^2$ x 10, $C_2^2.C_4^2$ x 8, $C_2^3:C_8$ x 6, $C_4^2:C_2^2$ x 6, $C_4^2:C_2^2$ x 6, $\OD_{16}:C_2^2$ x 4, $C_2^3.C_2^3$ x 4, $C_2^3.D_4$ x 4, $C_2^2:C_4^2$ x 4, $C_4^2:C_4$ x 4, $C_2^3:D_4$ x 4, $D_4:D_4$ x 4, $C_2^4\times C_4$ x 3, $D_4:D_4$ x 3, $C_4^2:C_2^2$ x 3, $C_2^2:\OD_{16}$ x 2, $C_2^3.D_4$ x 2, $D_4:C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3:Q_8$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.D_4$, $C_2^2.Q_{16}$, $C_2^2.D_8$, $C_8:C_2^3$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_4$, $C_2^3.D_4$, $C_2^2\times C_4^2$, $C_2^2:\SD_{16}$, $D_4:D_4$, $C_4^2:C_2^2$

Order 48: $C_2\times S_4$ x 25, $C_2^2\times A_4$ x 14, $C_2\times \SL(2,3)$ x 2, $A_4:C_4$ x 2, $\GL(2,3)$ x 2, $C_2^2\times D_6$, $C_2^2:A_4$, $C_4\times D_6$, $C_4\times A_4$, $C_2^2:C_{12}$, $C_6.D_4$, $D_6:C_4$

Order 32: $C_2^2\times D_4$ x 263, $C_2^3:C_4$ x 198, $C_2^2\wr C_2$ x 126, $C_2^5$ x 106, $C_2^3\times C_4$ x 39, $C_4:D_4$ x 38, $C_4\times D_4$ x 27, $C_2^3:C_4$ x 26, $C_2.C_4^2$ x 12, $C_4:D_4$ x 11, $C_2^2.D_4$ x 7, $C_4^2:C_2$ x 6, $\OD_{16}:C_2$ x 5, $C_2\times C_4^2$ x 5, $D_4:C_2^2$ x 4, $C_2^2.D_4$ x 4, $C_2^2:C_8$ x 3, $D_8:C_2$ x 3, $C_2^2:Q_8$ x 3, $D_4:C_2^2$ x 2, $C_2\times \OD_{16}$ x 2, $C_4^2:C_2$ x 2, $C_4\wr C_2$ x 2, $D_4:C_4$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_4^2:C_2$

Order 24: $C_2\times A_4$ x 15, $S_4$ x 10, $C_2\times D_6$ x 7, $\SL(2,3)$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $C_4\times S_3$, $C_2^2\times C_6$

Order 16: $C_2^4$ x 351, $C_2\times D_4$ x 310, $C_2^2:C_4$ x 126, $C_2^2\times C_4$ x 115, $C_4:C_4$ x 9, $C_4^2$ x 6, $\OD_{16}$ x 4, $D_4:C_2$ x 4, $\SD_{16}$ x 2, $D_8$ x 2, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2

Order 12: $D_6$ x 10, $C_2\times C_6$ x 4, $A_4$ x 4, $C_{12}$, $C_3:C_4$

Order 8: $C_2^3$ x 379, $C_2\times C_4$ x 89, $D_4$ x 83, $Q_8$ x 2, $C_8$ x 2

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

Order 4: $C_2^2$ x 130, $C_4$ x 15

Order 3: $C_3$

Order 2: $C_2$ x 16

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6144: $C_2^6.(C_4\times S_4)$ ($C_1$)

Order 3072: $C_2^7:S_4$ ($C_2$), $C_2^7.S_4$ ($C_2$), $C_2^6.(C_4\times A_4)$ ($C_2$)

Order 1536: $C_2^6.S_4$ ($C_4$), $C_2^6:S_4$ ($C_4$), $C_2^7:A_4$ ($C_2^2$)

Order 1024: $C_2^7.C_2^3$ ($S_3$)

Order 768: $C_2^6:A_4$ ($C_2\times C_4$)

Order 512: $C_2^7:C_2^2$ ($D_6$)

Order 256: $C_2^6:C_4$ ($S_4$), $C_4^2:C_2^4$ ($C_4\times S_3$)

Order 128: $C_2^7$ ($C_2\times S_4$)

Order 64: $C_2^6$ ($C_4\times S_4$)

Order 32: $C_2^5$ ($C_2^3:S_4$)

Order 16: $C_2^4$ ($(C_2^2\times C_4):S_4$), $C_2^4$ ($C_2\wr S_4$), $C_2^4$ ($C_2^5.D_6$)

Order 8: $C_2^3$ ($(C_2^3\times C_4):S_4$)

Order 4: $C_2^2$ ($C_2^6:(C_4\times S_3)$)

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

Order 1: $C_1$ ($C_2^6.(C_4\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6144: $C_2^6.(C_4\times S_4)$ ($C_1$)

Order 3072: $C_2^7:S_4$ ($C_2$), $C_2^7.S_4$ ($C_2$), $C_2^6.(C_4\times A_4)$ ($C_2$)

Order 1536: $C_2^6.S_4$ ($C_4$), $C_2^6:S_4$ ($C_4$), $C_2^7:A_4$ ($C_2^2$)

Order 1024: $C_2^7.C_2^3$ ($S_3$)

Order 768: $C_2^6:A_4$ ($C_2\times C_4$)

Order 512: $C_2^7:C_2^2$ ($D_6$)

Order 256: $C_2^6:C_4$ ($S_4$), $C_4^2:C_2^4$ ($C_4\times S_3$)

Order 128: $C_2^7$ ($C_2\times S_4$)

Order 64: $C_2^6$ ($C_4\times S_4$)

Order 32: $C_2^5$ ($C_2^3:S_4$)

Order 16: $C_2^4$ ($(C_2^2\times C_4):S_4$), $C_2^4$ ($C_2\wr S_4$), $C_2^4$ ($C_2^5.D_6$)

Order 8: $C_2^3$ ($(C_2^3\times C_4):S_4$)

Order 4: $C_2^2$ ($C_2^6:(C_4\times S_3)$)

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

Order 1: $C_1$ ($C_2^6.(C_4\times S_4)$)

Series

Derived series

$C_2^6.(C_4\times S_4)$

$\rhd$

$C_2^6:A_4$

$\rhd$

$C_4^2:C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^6.(C_4\times S_4)$

$\rhd$

$C_2^6.(C_4\times A_4)$

$\rhd$

$C_2^7:A_4$

$\rhd$

$C_2^6:A_4$

$\rhd$

$C_4^2:C_2^4$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6.(C_4\times S_4)$

$\rhd$

$C_2^6:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

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

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

Character theory

Complex character table

See the $63 \times 63$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $52 \times 52$ rational character table.