Abstract group 55296.cl: $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$

• Label: 55296.cl
• Order: \( 2^{11} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $20$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$
Order:	\(55296\)\(\medspace = 2^{11} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^8.C_3^3.C_2^6$, of order \(442368\)\(\medspace = 2^{14} \cdot 3^{3} \)
Composition factors:	$C_2$ x 11, $C_3$ x 3
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	991	2672	7200	21392	23040	55296
Conjugacy classes	1	30	11	23	95	48	208
Divisions	1	30	7	21	53	23	135
Autjugacy classes	1	27	7	15	46	13	109

Dimension	1	2	3	4	6	8	9	12	18	24	27	36	54	72	108
Irr. complex chars.	12	27	12	6	75	0	4	12	41	0	4	6	9	0	0	208
Irr. rational chars.	4	9	4	13	17	2	4	33	15	5	4	15	7	2	1	135

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid c^{6}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,3,6,5)(2,4)(7,9)(8,10)(11,12)(13,15,18,20,14,17)(16,19), (1,2)(3,5)(4,7)(6,8)(10,11)(13,14,16)(17,19,20)\rangle$
Transitive group:	36T16923				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(A_4^2:C_2^4)$ . $S_4$	$A_4^2$ . $(C_2^4:S_4)$	$(C_2^5.S_4^2)$ . $C_3$	$(C_2^9.C_3)$ . $S_3^2$	all 51

Elements of the group are displayed as permutations of degree 20.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 55 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^8.(C_3\times S_3^2)$
Commutator:	$G' \simeq$ $C_2^9:C_3^2$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^6.C_3^3.C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9$	$G/\operatorname{Fit} \simeq$ $C_3\times S_3^2$
Radical:	$R \simeq$ $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^7$	$G/\operatorname{soc} \simeq$ $C_6^2:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 96 (order at least 576) are shown, as well as all normal subgroups of any index.

Order 55296: $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$

Order 27648: $C_2^4:C_3.A_4^2.C_2^2$, $(C_2^3\times A_4^2).S_4$, $C_2^8.(C_3^2\times D_6)$

Order 18432: $C_2^4:C_3.C_2^4.C_6.C_2^2$, $C_2^5.S_4^2$, $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$

Order 13824: $C_2^8.C_3^3.C_2$, $C_2^4:C_3.A_4^2.C_2$, $A_4^2.\GL(2,\mathbb{Z}/4)$, $C_2^8:(S_3\times C_3^2)$

Order 9216: $C_2^5.(A_4\times S_4)$ x 2, $C_2^4.C_6^2.C_2^4$, $C_2^4.C_6^2.C_4^2$, $C_2^4.C_6^2.C_2^3.C_2$, $C_2^4.C_6^2.C_2^3.C_2$, $C_2^4.C_6^2.C_2^4$, $C_2^2.A_4^2.C_2^4$, $C_2^6.C_6^2.C_2^2$, $C_2^4.C_6^2.C_4.C_2^2$, $C_2^5.\PSOPlus(4,3)$, $C_2^5.(A_4\times S_4)$, $C_2^5.(A_4\times S_4)$, $C_2^6.(A_4\times D_6)$, $C_2^7.(C_3\times S_4)$, $C_2^6.A_4^2$

Order 6912: $A_4.A_4^2.C_2^2$ x 2, $A_4^3:C_4$ x 2, $C_2^2.A_4^2.C_6.C_2$, $C_2^8.C_3^3$, $A_4^3:C_2^2$

Order 6144: $C_2^4:C_3.C_2^5.C_2^2$, $C_2^4:C_3.C_2^6.C_2$, $C_2^8.C_6.C_2^2$

Order 4608: $C_2^4.(A_4\times S_4)$ x 6, $C_2^4.C_6^2.C_2^3$ x 5, $C_2^6.C_6^2.C_2$ x 4, $(C_2^2\times A_4)\wr C_2$ x 3, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^5.C_6^2.C_2^2$ x 2, $C_2^4:(A_4\times S_4)$ x 2, $C_2^9:C_3^2$ x 2, $C_2^9:C_3^2$ x 2, $C_2^6.(S_3\times A_4)$ x 2, $C_2^2\wr S_3:C_{12}$ x 2, $(C_2^3\times A_4^2):C_4$ x 2, $C_2^4.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^3$, $C_2^2\times A_4^2.C_2^3$, $C_2^5.C_6^2.C_2^2$, $C_2\times C_2^5.C_6^2.C_2$, $C_2^8.C_3.C_6$, $C_2^6.C_6^2.C_2$, $C_2^4.C_6^2.C_2^3$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^4:(A_4\times S_4)$, $A_4\times C_2^4:S_4$, $C_2^5:A_4^2$, $C_2^6:(C_3\times S_4)$

Order 3456: $C_2\times A_4^3$ x 3, $S_4\times A_4^2$ x 3, $C_2^4:(C_3^2\times S_4)$ x 2, $C_2^6.C_3^3.C_2$, $(C_3\times A_4^2):D_4$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $A_4^3:C_2$

Order 3072: $C_2^4:C_3.C_2^5.C_2$ x 2, $C_2^4:C_3.C_2^5.C_2$ x 2, $C_2^4:C_3.C_2^5.C_2$ x 2, $C_2^6.C_6.C_2^3$ x 2, $C_2^6.C_6.C_2^3$ x 2, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^6.C_6.C_2^3$, $C_2^4:C_3.C_2^5.C_2$, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^6.C_6.C_2^3$, $C_2^6.C_6.C_2^3$, $C_2^4:C_3.C_2^3.C_2^3$, $(C_2^4\times A_4).C_2^4$, $C_2^6.C_6.C_4.C_2$, $C_2^6.C_6.C_2^3$, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^9.C_6$, $C_2^7.S_4$, $C_2^7.S_4$

Order 2304: $A_4\times C_2^3:S_4$ x 15, $C_2^6:C_6^2$ x 14, $(C_2^2\times A_4^2):C_4$ x 13, $A_4^2.C_2^4$ x 8, $C_2\times A_4^2:D_4$ x 8, $C_2^5.(C_6\times C_{12})$ x 5, $A_4^2.C_2^3.C_2$ x 4, $C_2^3:A_4\times S_4$ x 4, $A_4^2:C_2^4$ x 3, $C_2^4:A_4^2$ x 3, $(C_2^3\times A_4).S_4$ x 3, $C_2^5.(C_3\times S_4)$ x 3, $C_2^4.C_6^2.C_2^2$ x 2, $C_2^5.C_6^2.C_2$ x 2, $C_2^4.C_6^2.C_2^2$ x 2, $C_2^4:A_4^2$ x 2, $C_2^4:A_4^2$ x 2, $(A_4\times C_2^4):C_{12}$ x 2, $C_2^4:A_4^2$ x 2, $A_4^2.C_4.C_2^2$, $A_4^2.C_2^2.C_2^2$, $C_2^4.C_6^2.C_2^2$, $C_2^8.C_3^2$, $A_4^2:C_2^4$, $C_2^4:(C_6\times S_4)$, $A_4\times C_2^3.S_4$, $C_2^4:(A_4\times D_6)$, $C_2^2\wr S_3\times C_6$

Order 2048: $C_2^9.C_2^2$

Order 1728: $C_6\times A_4\times S_4$ x 3, $A_4^3$ x 3, $C_2\times C_6^2:S_4$, $C_2^6:C_3^3$, $D_6\times A_4^2$, $A_4\wr C_2\times C_6$, $(C_2\times C_6^2).S_4$, $(C_3\times A_4^2):C_4$, $(C_2^3\times C_6^2):C_6$

Order 1536: $C_2^2\times C_2^3.(C_6\times D_4)$ x 24, $C_2^6:S_4$ x 8, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^8:C_6$ x 8, $C_2\times C_2^4.(C_6.D_4)$ x 7, $C_2^8.C_6$ x 6, $C_2^8:C_6$ x 5, $C_2\times C_2^4.D_6:C_4$ x 4, $C_2^8.C_6$ x 4, $C_2^8.C_6$ x 4, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^9.C_3$ x 3, $C_2^8.C_6$ x 3, $C_2^3\times C_2^2:C_4\times A_4$ x 3, $C_2^7.D_6$ x 2, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2\times C_2^4:C_3.D_4$ x 2, $C_2^9.C_3$ x 2, $C_2^8.C_6$ x 2, $C_2^8:C_6$ x 2, $C_2^7.C_2.C_6$ x 2, $C_2^5:(C_4\times A_4)$ x 2, $C_2^7:C_{12}$ x 2, $C_2^2\times C_2^4:C_3.D_4$, $C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^7.D_6$, $(C_2^6\times C_4).S_3$, $C_2^6.C_2.D_6$, $C_2^7.D_6$, $C_2^3\times C_2^4\times A_4$, $C_2^3\wr C_3$, $A_4\times C_2^4.C_2^3$, $C_2^5.C_2^4.C_3$, $C_2^7.C_2.C_6$, $C_2^7.C_2.C_6$

Order 1152: $C_2^3\times A_4^2$ x 34, $C_2^3:A_4^2$ x 26, $A_4^2:C_2^3$ x 18, $D_4\times A_4^2$ x 12, $A_4\times C_2^2:S_4$ x 11, $C_2\times A_4^2:C_4$ x 10, $C_2^5:C_6^2$ x 8, $C_2^3:A_4^2$ x 7, $C_2^5.C_6^2$ x 7, $C_3\times C_2^2\wr S_3$ x 6, $C_2^2:A_4\times S_4$ x 6, $A_4^2:D_4$ x 6, $A_4^2:C_2^3$ x 3, $C_2^5:C_6^2$ x 3, $C_6\times C_2^2:C_4\times A_4$ x 3, $C_2^4:(S_3\times A_4)$ x 2, $C_2^3:A_4^2$ x 2, $(C_6^2\times D_4).C_2^2$, $C_2^5.S_3^2$, $A_4\times \GL(2,\mathbb{Z}/4)$, $A_4:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times S_4):A_4$, $C_2^4:(S_3\times A_4)$, $C_3\times C_2^4:S_4$, $C_2^5.S_3^2$, $C_2^4\times C_6\times A_4$, $C_2^5:C_6^2$, $C_6\times C_2^6:C_3$

Order 1024: $C_2^8.C_2^2$ x 4, $C_2^7:D_4$ x 4, $C_2^8.C_2^2$, $C_2^8.C_2^2$, $C_2^7.C_2^3$, $C_2^9.C_2$, $C_2^6.C_2^4$, $C_2^8:C_4$, $C_2^9.C_2$

Order 864: $A_4^2:C_6$ x 6, $C_6\times A_4^2$ x 5, $C_2^5:C_3^3$, $C_6^2:S_4$, $S_3\times A_4^2$, $A_4^2:C_6$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $(C_6\times D_6):A_4$

Order 768: $C_2^7:C_6$ x 156, $C_2^7:C_6$ x 60, $C_2^5:S_4$ x 48, $C_2^7.C_6$ x 36, $C_2^7:C_6$ x 28, $C_2^8.C_3$ x 21, $C_2^8.C_3$ x 20, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 17, $C_2^6:C_{12}$ x 15, $C_2^6:C_{12}$ x 14, $C_2^7:C_6$ x 14, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^5:S_4$ x 8, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^5.S_4$ x 7, $C_2^2\times C_2^2:(A_4:C_4)$ x 6, $C_2^7.C_6$ x 5, $C_2^6:C_{12}$ x 4, $A_4.(C_2^5.C_2)$ x 3, $C_2^6:A_4$ x 3, $C_2\times C_4\times C_2^3\times A_4$ x 3, $C_2^6:A_4$ x 3, $S_4\times C_2^5$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.(C_6.Q_8)$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $A_4.C_2^5.C_2$ x 2, $C_2^6:A_4$ x 2, $C_2^6:A_4$ x 2, $C_2^6:C_{12}$ x 2, $C_2^5.(C_2\times C_{12})$ x 2, $C_2^6.C_2^2.C_3$ x 2, $C_2^7:C_6$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $A_4.C_2^4:C_2^2$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $A_4.C_2^2:C_4:C_4$, $C_2^6.C_2.C_6$, $C_2^7.C_6$, $C_2^6.C_2.C_6$

Order 576: $C_2^2\times A_4^2$ x 44, $A_4^2:C_2^2$ x 41, $C_2^2:A_4^2$ x 24, $C_2^4:C_6^2$ x 24, $C_2^2:S_4\times C_6$ x 13, $C_2^4.C_6^2$ x 12, $C_2^4:C_6^2$ x 10, $C_2^2:A_4^2$ x 7, $A_4^2:C_4$ x 5, $A_4^2:C_2^2$ x 4, $C_2^4:C_6^2$ x 3, $C_2^2\times C_6\times S_4$ x 3, $C_2^4.C_6^2$ x 3, $C_4\times A_4^2$ x 3, $C_2^2:A_4\times D_6$ x 2, $(C_2^2\times C_6).S_4$ x 2, $C_3\times C_2^3.S_4$ x 2, $A_4^2:C_4$ x 2, $A_4^2:C_4$ x 2, $C_2^4:C_6^2$ x 2, $C_2^4:(C_6\times S_3)$ x 2, $C_2^6:C_3^2$, $C_2^2\times A_4\times D_6$, $C_2^4:C_6^2$, $C_6^2.C_2^4$, $C_6^2.C_2^4$, $(C_2^3\times C_6):C_{12}$, $(C_6\times D_4):C_{12}$, $(C_2^2\times D_6):C_{12}$, $(C_2^3\times C_6):C_{12}$, $C_6^2.C_4^2$

Order 512: $C_2^9$

Order 384: $C_2^5:A_4$ x 2

Order 288: $C_2\times A_4^2$

Order 256: $C_2^8$

Order 192: $C_2^2\wr C_3$ x 2, $C_2^3:S_4$

Order 144: $A_4^2$

Order 128: $C_2^7$

Order 96: $C_2^3:A_4$ x 3

Order 64: $C_2^6$

Order 48: $C_2^2:A_4$ x 3

Order 32: $C_2^5$ x 2

Order 27: $C_3^3$

Order 16: $C_2^4$ x 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 96 (order at least 576) are shown, as well as all normal subgroups of any index.

Order 55296: $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$

Order 27648: $C_2^4:C_3.A_4^2.C_2^2$, $(C_2^3\times A_4^2).S_4$, $C_2^8.(C_3^2\times D_6)$

Order 18432: $C_2^4:C_3.C_2^4.C_6.C_2^2$, $C_2^5.S_4^2$, $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$

Order 13824: $C_2^8.C_3^3.C_2$, $C_2^4:C_3.A_4^2.C_2$, $A_4^2.\GL(2,\mathbb{Z}/4)$, $C_2^8:(S_3\times C_3^2)$

Order 9216: $C_2^5.(A_4\times S_4)$ x 2, $C_2^4.C_6^2.C_2^4$, $C_2^4.C_6^2.C_4^2$, $C_2^4.C_6^2.C_2^3.C_2$, $C_2^4.C_6^2.C_2^3.C_2$, $C_2^4.C_6^2.C_2^4$, $C_2^2.A_4^2.C_2^4$, $C_2^6.C_6^2.C_2^2$, $C_2^4.C_6^2.C_4.C_2^2$, $C_2^5.\PSOPlus(4,3)$, $C_2^5.(A_4\times S_4)$, $C_2^5.(A_4\times S_4)$, $C_2^6.(A_4\times D_6)$, $C_2^7.(C_3\times S_4)$, $C_2^6.A_4^2$

Order 6912: $A_4.A_4^2.C_2^2$ x 2, $A_4^3:C_4$ x 2, $C_2^2.A_4^2.C_6.C_2$, $C_2^8.C_3^3$, $A_4^3:C_2^2$

Order 6144: $C_2^4:C_3.C_2^5.C_2^2$, $C_2^4:C_3.C_2^6.C_2$, $C_2^8.C_6.C_2^2$

Order 4608: $C_2^4.C_6^2.C_2^3$ x 5, $C_2^4.(A_4\times S_4)$ x 4, $C_2^6.C_6^2.C_2$ x 3, $(C_2^2\times A_4)\wr C_2$ x 3, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^5.C_6^2.C_2^2$ x 2, $C_2^4:(A_4\times S_4)$ x 2, $C_2^9:C_3^2$ x 2, $C_2^9:C_3^2$ x 2, $C_2^4.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^3$, $C_2^2\times A_4^2.C_2^3$, $C_2^5.C_6^2.C_2^2$, $C_2\times C_2^5.C_6^2.C_2$, $C_2^8.C_3.C_6$, $C_2^6.C_6^2.C_2$, $C_2^4.C_6^2.C_2^3$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^4:(A_4\times S_4)$, $A_4\times C_2^4:S_4$, $C_2^5:A_4^2$, $C_2^6:(C_3\times S_4)$, $C_2^6.(S_3\times A_4)$, $C_2^2\wr S_3:C_{12}$, $(C_2^3\times A_4^2):C_4$

Order 3456: $C_2\times A_4^3$ x 3, $S_4\times A_4^2$ x 2, $C_2^6.C_3^3.C_2$, $C_2^4:(C_3^2\times S_4)$, $(C_3\times A_4^2):D_4$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $A_4^3:C_2$

Order 3072: $C_2^4:C_3.C_2^5.C_2$ x 2, $C_2^4:C_3.C_2^5.C_2$ x 2, $C_2^4:C_3.C_2^5.C_2$ x 2, $C_2^6.C_6.C_2^3$ x 2, $C_2^6.C_6.C_2^3$ x 2, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^6.C_6.C_2^3$, $C_2^4:C_3.C_2^5.C_2$, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^6.C_6.C_2^3$, $C_2^6.C_6.C_2^3$, $C_2^4:C_3.C_2^3.C_2^3$, $(C_2^4\times A_4).C_2^4$, $C_2^6.C_6.C_4.C_2$, $C_2^6.C_6.C_2^3$, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^9.C_6$, $C_2^7.S_4$, $C_2^7.S_4$

Order 2304: $C_2^6:C_6^2$ x 12, $(C_2^2\times A_4^2):C_4$ x 12, $A_4\times C_2^3:S_4$ x 11, $A_4^2.C_2^4$ x 8, $C_2\times A_4^2:D_4$ x 8, $A_4^2.C_2^3.C_2$ x 4, $C_2^3:A_4\times S_4$ x 4, $C_2^5.(C_6\times C_{12})$ x 3, $C_2^4:A_4^2$ x 3, $(C_2^3\times A_4).S_4$ x 3, $C_2^4.C_6^2.C_2^2$ x 2, $C_2^4.C_6^2.C_2^2$ x 2, $A_4^2:C_2^4$ x 2, $C_2^4:A_4^2$ x 2, $C_2^4:A_4^2$ x 2, $(A_4\times C_2^4):C_{12}$ x 2, $C_2^5.(C_3\times S_4)$ x 2, $C_2^4:A_4^2$ x 2, $A_4^2.C_4.C_2^2$, $A_4^2.C_2^2.C_2^2$, $C_2^4.C_6^2.C_2^2$, $C_2^8.C_3^2$, $C_2^5.C_6^2.C_2$, $A_4^2:C_2^4$, $C_2^4:(C_6\times S_4)$, $A_4\times C_2^3.S_4$, $C_2^4:(A_4\times D_6)$, $C_2^2\wr S_3\times C_6$

Order 2048: $C_2^9.C_2^2$

Order 1728: $A_4^3$ x 3, $C_6\times A_4\times S_4$ x 2, $C_2\times C_6^2:S_4$, $C_2^6:C_3^3$, $D_6\times A_4^2$, $A_4\wr C_2\times C_6$, $(C_2\times C_6^2).S_4$, $(C_3\times A_4^2):C_4$, $(C_2^3\times C_6^2):C_6$

Order 1536: $C_2^2\times C_2^3.(C_6\times D_4)$ x 12, $C_2^6:S_4$ x 7, $C_2\times C_2^4.(C_6.D_4)$ x 7, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^8:C_6$ x 6, $C_2^8.C_6$ x 6, $C_2^8:C_6$ x 5, $C_2^8.C_6$ x 4, $C_2^8.C_6$ x 4, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 3, $C_2\times C_2^4.D_6:C_4$ x 3, $C_2^9.C_3$ x 3, $C_2^7.D_6$ x 2, $C_2^2\times C_2^4:C_3.D_4$ x 2, $C_2^9.C_3$ x 2, $C_2^8.C_6$ x 2, $C_2^3\times C_2^2:C_4\times A_4$ x 2, $C_2^8.C_6$ x 2, $C_2^7.C_2.C_6$ x 2, $C_2^2\times C_2^4:C_3.D_4$, $C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^7.D_6$, $(C_2^6\times C_4).S_3$, $C_2^4.\GL(2,\mathbb{Z}/4)$, $C_2^4.\GL(2,\mathbb{Z}/4)$, $C_2^6.C_2.D_6$, $C_2^7.D_6$, $C_2^3\times C_2^4\times A_4$, $C_2^3\wr C_3$, $A_4\times C_2^4.C_2^3$, $C_2^5.C_2^4.C_3$, $C_2^8:C_6$, $C_2^7.C_2.C_6$, $C_2^7.C_2.C_6$, $C_2^5:(C_4\times A_4)$, $C_2^7:C_{12}$

Order 1152: $C_2^3\times A_4^2$ x 33, $C_2^3:A_4^2$ x 22, $C_2\times A_4^2:C_4$ x 10, $A_4^2:C_2^3$ x 8, $D_4\times A_4^2$ x 8, $A_4\times C_2^2:S_4$ x 7, $C_2^3:A_4^2$ x 6, $C_2^5.C_6^2$ x 6, $A_4^2:D_4$ x 6, $C_3\times C_2^2\wr S_3$ x 4, $C_2^2:A_4\times S_4$ x 4, $A_4^2:C_2^3$ x 3, $C_2^5:C_6^2$ x 3, $C_2^3:A_4^2$ x 2, $C_2^5:C_6^2$ x 2, $C_6\times C_2^2:C_4\times A_4$ x 2, $(C_6^2\times D_4).C_2^2$, $C_2^4:(S_3\times A_4)$, $C_2^5.S_3^2$, $A_4\times \GL(2,\mathbb{Z}/4)$, $A_4:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times S_4):A_4$, $C_2^4:(S_3\times A_4)$, $C_3\times C_2^4:S_4$, $C_2^5.S_3^2$, $C_2^4\times C_6\times A_4$, $C_2^5:C_6^2$, $C_6\times C_2^6:C_3$

Order 1024: $C_2^8.C_2^2$ x 4, $C_2^7:D_4$ x 4, $C_2^8.C_2^2$, $C_2^8.C_2^2$, $C_2^7.C_2^3$, $C_2^9.C_2$, $C_2^6.C_2^4$, $C_2^8:C_4$, $C_2^9.C_2$

Order 864: $C_6\times A_4^2$ x 4, $A_4^2:C_6$ x 2, $C_2^5:C_3^3$, $C_6^2:S_4$, $S_3\times A_4^2$, $A_4^2:C_6$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $(C_6\times D_6):A_4$

Order 768: $C_2^7:C_6$ x 51, $C_2^7:C_6$ x 32, $C_2^5:S_4$ x 31, $C_2^7:C_6$ x 24, $C_2^8.C_3$ x 21, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 16, $C_2^8.C_3$ x 16, $C_2^7.C_6$ x 15, $C_2^7:C_6$ x 14, $C_2^6:C_{12}$ x 14, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^6:C_{12}$ x 8, $C_2^2\times C_2^2:(A_4:C_4)$ x 6, $C_2^5.S_4$ x 5, $C_2^5:S_4$ x 5, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 5, $C_2^7.C_6$ x 5, $C_2^6:C_{12}$ x 4, $A_4.(C_2^5.C_2)$ x 3, $C_2^6:A_4$ x 3, $C_2^6:A_4$ x 3, $S_4\times C_2^5$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.(C_6.Q_8)$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $A_4.C_2^5.C_2$ x 2, $C_2^6:A_4$ x 2, $C_2\times C_4\times C_2^3\times A_4$ x 2, $C_2^6:A_4$ x 2, $C_2^6:C_{12}$ x 2, $C_2^5.(C_2\times C_{12})$ x 2, $C_2^6.C_2^2.C_3$ x 2, $C_2^7:C_6$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $A_4.C_2^4:C_2^2$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $A_4.C_2^2:C_4:C_4$, $C_2^6.C_2.C_6$, $C_2^7.C_6$, $C_2^6.C_2.C_6$

Order 576: $C_2^2\times A_4^2$ x 40, $A_4^2:C_2^2$ x 21, $C_2^2:A_4^2$ x 20, $C_2^4:C_6^2$ x 12, $C_2^2:S_4\times C_6$ x 9, $C_2^4:C_6^2$ x 8, $C_2^2:A_4^2$ x 6, $A_4^2:C_4$ x 5, $A_4^2:C_2^2$ x 4, $C_2^4.C_6^2$ x 4, $C_2^4:C_6^2$ x 3, $C_2^2:A_4\times D_6$ x 2, $C_2^2\times C_6\times S_4$ x 2, $C_2^4.C_6^2$ x 2, $(C_2^2\times C_6).S_4$ x 2, $C_3\times C_2^3.S_4$ x 2, $C_4\times A_4^2$ x 2, $A_4^2:C_4$ x 2, $A_4^2:C_4$ x 2, $C_2^4:C_6^2$ x 2, $C_2^4:(C_6\times S_3)$ x 2, $C_2^6:C_3^2$, $C_2^2\times A_4\times D_6$, $C_2^4:C_6^2$, $C_6^2.C_2^4$, $C_6^2.C_2^4$, $(C_2^3\times C_6):C_{12}$, $(C_6\times D_4):C_{12}$, $(C_2^2\times D_6):C_{12}$, $(C_2^3\times C_6):C_{12}$, $C_6^2.C_4^2$

Order 512: $C_2^9$

Order 384: $C_2^5:A_4$ x 2

Order 288: $C_2\times A_4^2$

Order 256: $C_2^8$

Order 192: $C_2^2\wr C_3$ x 2, $C_2^3:S_4$

Order 144: $A_4^2$

Order 128: $C_2^7$

Order 96: $C_2^3:A_4$ x 3

Order 64: $C_2^6$

Order 48: $C_2^2:A_4$ x 3

Order 32: $C_2^5$ x 2

Order 27: $C_3^3$

Order 16: $C_2^4$ x 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 55296: $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 27648: $C_2^4:C_3.A_4^2.C_2^2$ ($C_2$), $(C_2^3\times A_4^2).S_4$ ($C_2$), $C_2^8.(C_3^2\times D_6)$ ($C_2$)

Order 18432: $C_2^5.S_4^2$ ($C_3$)

Order 13824: $C_2^8.C_3^3.C_2$ ($C_2^2$)

Order 9216: $C_2^4.C_6^2.C_2^4$ ($S_3$), $C_2^5.(A_4\times S_4)$ ($C_6$), $C_2^5.(A_4\times S_4)$ ($S_3$), $C_2^5.\PSOPlus(4,3)$ ($C_6$), $C_2^5.(A_4\times S_4)$ ($C_6$)

Order 6912: $C_2^8.C_3^3$ ($D_4$)

Order 4608: $C_2^2\times A_4^2.C_2^3$ ($D_6$), $C_2^9:C_3^2$ ($C_2\times C_6$), $C_2^9:C_3^2$ ($D_6$)

Order 3072: $C_2^4:C_3.C_2^5.C_2$ ($C_3\times S_3$), $C_2^7.S_4$ ($C_3\times S_3$)

Order 2304: $A_4^2.C_2^4$ ($C_3:D_4$), $A_4^2:C_2^4$ ($S_4$), $C_2^4:A_4^2$ ($C_3:D_4$), $C_2^4:A_4^2$ ($C_3\times D_4$)

Order 1536: $C_2^9.C_3$ ($C_6\times S_3$) x 2, $C_2^9.C_3$ ($S_3^2$)

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

Order 768: $C_2^8.C_3$ ($C_6\wr C_2$) x 2, $C_2^5:S_4$ ($C_3\times S_4$), $C_2^8.C_3$ ($D_6:S_3$)

Order 576: $C_2^2\times A_4^2$ ($\GL(2,\mathbb{Z}/4)$)

Order 512: $C_2^9$ ($C_3\times S_3^2$)

Order 384: $C_2^5:A_4$ ($C_6\times S_4$), $C_2^5:A_4$ ($S_3\times S_4$)

Order 288: $C_2\times A_4^2$ ($C_2^3:S_4$)

Order 256: $C_2^8$ ($C_3^3:D_4$)

Order 192: $C_2^2\wr C_3$ ($C_3\times \GL(2,\mathbb{Z}/4)$), $C_2^2\wr C_3$ ($D_6:S_4$), $C_2^3:S_4$ ($A_4\wr C_2$)

Order 144: $A_4^2$ ($C_2^4:S_4$)

Order 128: $C_2^7$ ($C_6^2:D_6$)

Order 96: $C_2^3:A_4$ ($A_4^2:C_2^2$), $C_2^3:A_4$ ($C_3\times C_2^3:S_4$), $C_2^3:A_4$ ($C_2^4:S_3^2$)

Order 64: $C_2^6$ ($C_3^2:\GL(2,\mathbb{Z}/4)$)

Order 48: $C_2^2:A_4$ ($A_4^2:D_4$), $C_2^2:A_4$ ($C_3\times C_2^4:S_4$), $C_2^2:A_4$ ($C_2^5.S_3^2$)

Order 32: $C_2^5$ ($A_4^2:D_6$), $C_2^5$ ($C_3\times C_2^4:S_3^2$)

Order 16: $C_2^4$ ($(C_3\times A_4^2):D_4$), $C_2^4$ ($C_6^2.\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_2^6.C_3^3.C_2^2$)

Order 4: $C_2^2$ ($A_4^2.\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_2^8.(C_3\times S_3^2)$)

Order 1: $C_1$ ($(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 55296: $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 27648: $C_2^4:C_3.A_4^2.C_2^2$ ($C_2$), $(C_2^3\times A_4^2).S_4$ ($C_2$), $C_2^8.(C_3^2\times D_6)$ ($C_2$)

Order 18432: $C_2^5.S_4^2$ ($C_3$)

Order 13824: $C_2^8.C_3^3.C_2$ ($C_2^2$)

Order 9216: $C_2^4.C_6^2.C_2^4$ ($S_3$), $C_2^5.(A_4\times S_4)$ ($C_6$), $C_2^5.(A_4\times S_4)$ ($S_3$), $C_2^5.\PSOPlus(4,3)$ ($C_6$), $C_2^5.(A_4\times S_4)$ ($C_6$)

Order 6912: $C_2^8.C_3^3$ ($D_4$)

Order 4608: $C_2^2\times A_4^2.C_2^3$ ($D_6$), $C_2^9:C_3^2$ ($C_2\times C_6$), $C_2^9:C_3^2$ ($D_6$)

Order 3072: $C_2^4:C_3.C_2^5.C_2$ ($C_3\times S_3$), $C_2^7.S_4$ ($C_3\times S_3$)

Order 2304: $A_4^2.C_2^4$ ($C_3:D_4$), $A_4^2:C_2^4$ ($S_4$), $C_2^4:A_4^2$ ($C_3:D_4$), $C_2^4:A_4^2$ ($C_3\times D_4$)

Order 1536: $C_2^9.C_3$ ($C_6\times S_3$) x 2, $C_2^9.C_3$ ($S_3^2$)

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

Order 768: $C_2^8.C_3$ ($C_6\wr C_2$) x 2, $C_2^5:S_4$ ($C_3\times S_4$), $C_2^8.C_3$ ($D_6:S_3$)

Order 576: $C_2^2\times A_4^2$ ($\GL(2,\mathbb{Z}/4)$)

Order 512: $C_2^9$ ($C_3\times S_3^2$)

Order 384: $C_2^5:A_4$ ($C_6\times S_4$), $C_2^5:A_4$ ($S_3\times S_4$)

Order 288: $C_2\times A_4^2$ ($C_2^3:S_4$)

Order 256: $C_2^8$ ($C_3^3:D_4$)

Order 192: $C_2^2\wr C_3$ ($C_3\times \GL(2,\mathbb{Z}/4)$), $C_2^2\wr C_3$ ($D_6:S_4$), $C_2^3:S_4$ ($A_4\wr C_2$)

Order 144: $A_4^2$ ($C_2^4:S_4$)

Order 128: $C_2^7$ ($C_6^2:D_6$)

Order 96: $C_2^3:A_4$ ($A_4^2:C_2^2$), $C_2^3:A_4$ ($C_3\times C_2^3:S_4$), $C_2^3:A_4$ ($C_2^4:S_3^2$)

Order 64: $C_2^6$ ($C_3^2:\GL(2,\mathbb{Z}/4)$)

Order 48: $C_2^2:A_4$ ($A_4^2:D_4$), $C_2^2:A_4$ ($C_3\times C_2^4:S_4$), $C_2^2:A_4$ ($C_2^5.S_3^2$)

Order 32: $C_2^5$ ($A_4^2:D_6$), $C_2^5$ ($C_3\times C_2^4:S_3^2$)

Order 16: $C_2^4$ ($(C_3\times A_4^2):D_4$), $C_2^4$ ($C_6^2.\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($C_2^6.C_3^3.C_2^2$)

Order 4: $C_2^2$ ($A_4^2.\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_2^8.(C_3\times S_3^2)$)

Order 1: $C_1$ ($(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$)

Series

Derived series

$(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^9:C_3^2$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^4:C_3.A_4^2.C_2^2$

$\rhd$

$C_2^8.C_3^3.C_2$

$\rhd$

$C_2^9:C_3^2$

$\rhd$

$C_2^4:A_4^2$

$\rhd$

$C_2^8.C_3$

$\rhd$

$C_2^8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^9:C_3^2$

$\rhd$

$C_2^4:A_4^2$

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

See the $208 \times 208$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $135 \times 135$ rational character table (warning: may be slow to load).