Abstract group 165888.t: $A_4^2\wr C_2.C_4$

• Label: 165888.t
• Order: \( 2^{11} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $20$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$A_4^2\wr C_2.C_4$
Order:	\(165888\)\(\medspace = 2^{11} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^8.C_3^4.C_2^5.C_2$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \)
Composition factors:	$C_2$ x 11, $C_3$ x 4
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	799	6560	13536	57440	87552	165888
Conjugacy classes	1	15	26	16	122	32	212
Divisions	1	15	16	10	68	10	120
Autjugacy classes	1	12	12	5	47	5	82

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid a^{12}=b^{6}=c^{3}=d^{6}=e^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,2,4,9,8,14)(3,7,10,15)(5,11,6,13)(12,16)(17,18,19,20), (1,3,4,10,16,13) \!\cdots\! \rangle$
Transitive group:	24T18362	36T23316			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:	$(A_4^2\wr C_2)$ . $C_4$	$(C_2^9:C_3^2)$ . $S_3^2$ (2)	$C_2^9$ . $(C_3\wr C_2^2)$	$(C_2^4.A_4^2:D_6)$ . $S_3$	all 36

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

Homology

Abelianization:	$C_{2} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 12224084 subgroups in 41533 conjugacy classes, 56 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^8.C_3\wr C_2^2$
Commutator:	$G' \simeq$ $C_2^2.A_4^3$	$G/G' \simeq$ $C_2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^8.C_3\wr C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9$	$G/\operatorname{Fit} \simeq$ $C_3\wr C_2^2$
Radical:	$R \simeq$ $A_4^2\wr C_2.C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^9$	$G/\operatorname{soc} \simeq$ $C_3\wr C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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 384 (order at least 432) are shown, as well as all normal subgroups of any index.

Order 165888: $A_4^2\wr C_2.C_4$

Order 82944: $C_2.A_4^2\wr C_2$ x 2, $C_2\times A_4^2\wr C_2$

Order 55296: $C_2^9.(C_3\times S_3^2)$ x 2, $C_2^9.C_3:S_3^2$, $C_2^9.(C_3\times S_3^2)$

Order 41472: $A_4^2\wr C_2$ x 2, $C_2\times C_2^8.C_3^4$

Order 27648: $C_2^4:A_4^2.C_{12}$ x 4, $C_2^9.(S_3\times C_3^2)$ x 4, $C_2^4.A_4^2:D_6$ x 2, $(C_2^3\times A_4^2).S_4$ x 2, $C_2^8.(C_3^2\times D_6)$ x 2, $C_2^9.(S_3\times C_3^2)$ x 2, $(C_2^3\times A_4^2).S_4$, $C_2^8.(C_3^2\times D_6)$

Order 20736: $C_2^8.C_3^4$

Order 18432: $C_2^9.S_3^2$ x 2, $C_2^8.(S_3\times C_{12})$ x 2, $C_2^9.(C_6\times S_3)$, $C_2^9.S_3^2$

Order 13824: $C_2\times C_2^8.C_3^3$ x 6, $C_2\times C_2^8.C_3^3$ x 5, $C_2\times C_2^8.C_3^3$ x 4, $C_2^4:A_4^2.C_6$ x 4, $C_2^8.(S_3\times C_3^2)$ x 4, $(C_2^2\times A_4^2).S_4$ x 2, $C_2^8:(S_3\times C_3^2)$ x 2, $C_2^6.C_6^3$

Order 10368: $C_2\times C_2^6.C_3^4$

Order 9216: $C_2^9.(C_3\times S_3)$ x 8, $C_2^7.(S_3\times A_4)$ x 4, $C_2^7.(C_3\times S_4)$ x 4, $C_2^8.(C_6\times S_3)$ x 4, $C_2^4.C_6^2.C_2^3.C_2$ x 2, $C_2^5.\PSOPlus(4,3)$ x 2, $C_2^5.(A_4\times S_4)$ x 2, $C_2^6.(A_4\times D_6)$ x 2, $C_2^6.(C_6\times S_4)$ x 2, $C_2^6.(C_{12}\times A_4)$ x 2, $C_2\times C_2^4.C_6^2.C_2^3$, $C_2^5.\PSOPlus(4,3)$, $C_2^5.(A_4\times S_4)$, $C_2^6.A_4^2$

Order 6912: $C_2^8.C_3^3$ x 6, $C_2^2.A_4^3$ x 5, $C_2^2.A_4^3$ x 4, $C_2^6.C_3^3.C_2^2$ x 3, $A_4^3:C_4$ x 2, $A_4^3:C_4$ x 2, $A_4^3:C_2^2$, $A_4^3:C_2^2$

Order 6144: $C_2^9.D_6$ x 2, $C_2^9.D_6$, $C_2^8.(C_2\times C_{12})$

Order 5184: $(C_3\times A_4^2):C_{12}$ x 2, $C_3\times A_4^3$, $C_6\times A_4^2:S_3$

Order 4608: $C_2^9:C_3^2$ x 18, $C_2^9:C_3^2$ x 9, $C_2^8:(C_3\times S_3)$ x 8, $C_2^6.C_6^2.C_2$ x 5, $C_2\times C_2^8.C_3^2$ x 4, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^6.(C_3\times S_4)$ x 4, $C_2^6.(S_3\times A_4)$ x 4, $(C_2^2\times A_4)\wr C_2$ x 4, $C_2^6.C_6^2.C_2$ x 3, $A_4^2.C_2^5$ x 2, $C_2^4:\PSOPlus(4,3)$ x 2, $C_2^5:A_4^2$ x 2, $C_2^6:(C_3\times S_4)$ x 2, $C_2^9:C_3^2$ x 2, $C_2\times A_4^2.C_2^4$, $C_2\times C_2^6.C_6^2$, $C_2\times C_2^6.C_6^2$, $C_2\times C_2^4.C_6^2.C_2^2$, $C_2\times C_2^4.C_6^2.C_2^2$

Order 3456: $C_2\times A_4^3$ x 35, $C_2\times C_2^6.C_3^3$ x 4, $C_2^4.C_6^3$ x 3, $C_6\times C_2^6.C_3^2$ x 3, $C_6\times C_2^6.C_3^2$ x 2, $A_4^2:S_4$ x 2, $A_4^3:C_2$ x 2, $C_2\times C_2^6.C_3^3$

Order 3072: $C_2^8.C_{12}$ x 4, $C_2^7.S_4$ x 4, $C_2^4:C_3.C_2^5.C_2$ x 2, $C_2^6.C_6.C_2^3$ x 2, $C_2^7.S_4$ x 2, $C_2^9.C_6$ x 2, $C_2\times C_2^4.C_2^4:S_3$, $C_2\times C_2^6.C_6.C_2^2$

Order 2592: $C_6\times C_3\times A_4^2$ x 2, $(C_3\times A_4)\wr C_2$ x 2, $C_3\times C_6\times A_4^2$

Order 2304: $C_2^4:A_4^2$ x 18, $C_2^4:A_4^2$ x 18, $A_4^2.C_2^4$ x 13, $C_2^4:A_4^2$ x 12, $(C_2^2\times A_4^2):C_4$ x 12, $C_2^4:A_4^2$ x 9, $C_2\times A_4^2:D_4$ x 8, $A_4^2.C_2^2.C_2^2$ x 4, $C_2^8.C_3^2$ x 4, $C_2^2\wr C_3:C_{12}$ x 4, $C_2^5.(C_3\times S_4)$ x 4, $C_2\times A_4^2.C_2^3$ x 2, $(C_2^3\times A_4).S_4$ x 2, $(A_4\times C_2^4):C_{12}$ x 2, $A_4\times C_2^3.S_4$ x 2, $A_4\times C_2^4:C_{12}$ x 2, $C_2^4:A_4^2$ x 2, $C_2^4:(C_6\times S_4)$ x 2, $C_2^4:(A_4\times D_6)$ x 2, $C_2\times A_4^2.C_2^2.C_2$, $A_4^2:C_2^4$, $(C_2^3\times A_4):S_4$, $C_2^4:(C_6\times S_4)$, $A_4\times C_2^3:S_4$, $C_2^4:A_4^2$

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

Order 1728: $A_4^3$ x 33, $C_2^6:C_3^3$ x 9, $C_2^6:C_3^3$ x 6, $(C_3\times A_4^2):C_4$ x 6, $A_4^2:C_{12}$ x 6, $C_2^6:C_3^3$ x 4, $A_4^2:D_6$ x 3, $A_4\wr C_2\times C_6$ x 3, $(C_2\times C_6^2).S_4$ x 2, $(C_2^2\times C_6^2):C_{12}$ x 2, $C_2\times C_6^2:S_4$, $(C_2^3\times C_6^2):C_6$

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

Order 1296: $C_3^2\times A_4^2$ x 3

Order 1152: $C_2^3:A_4^2$ x 94, $C_2^3\times A_4^2$ x 75, $C_2^3:A_4^2$ x 62, $C_2\times A_4^2:C_4$ x 13, $A_4^2:D_4$ x 8, $C_2^5:C_6^2$ x 6, $C_2^4:(S_3\times A_4)$ x 4, $C_2^4:(C_3\times S_4)$ x 4, $A_4^2:C_2^3$ x 4, $C_6\times C_2^6:C_3$ x 4, $C_2^4\times C_6\times A_4$ x 3, $(C_2^2\times A_4):S_4$ x 2, $A_4\times C_2^2:S_4$ x 2, $A_4\wr C_2\times C_4$ x 2, $C_2^5.S_3^2$ x 2, $C_2^3:A_4^2$ x 2, $(C_2^2\times S_4):A_4$ x 2, $C_2^5.S_3^2$

Order 1024: $C_2^9.C_2$ x 2, $C_2^8.C_2^2$ x 2, $C_2^8.C_2^2$, $C_2^8.C_2^2$, $C_2^9.C_2$

Order 864: $C_6\times A_4^2$ x 75, $C_2^5:C_3^3$ x 6, $A_4^2:S_3$ x 6, $A_4^2:C_6$ x 6, $C_2^2:C_6^3$ x 3, $C_6^2:S_4$ x 2, $(C_6\times D_6):A_4$ x 2

Order 768: $C_2^8.C_3$ x 30, $C_2^8.C_3$ x 15, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 12, $C_2^6:A_4$ x 12, $C_2^6:C_{12}$ x 12, $C_2^5.S_4$ x 8, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^7:C_6$ x 8, $C_2^6:C_{12}$ x 8, $C_2^2\times C_2^2:(A_4:C_4)$ x 5, $C_2^6:A_4$ x 5, $C_2^7.C_6$ x 5, $C_2^5:S_4$ x 4, $C_2^6:A_4$ x 4, $A_4.(C_2^5.C_2)$ x 2, $C_2^7.C_6$ x 2, $C_2^5:S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^7:C_6$, $C_2^6:A_4$

Order 648: $C_6^3:C_3$, $C_3\times C_6.S_3^2$

Order 576: $C_2^2\times A_4^2$ x 136, $C_2^2:A_4^2$ x 82, $C_2^2:A_4^2$ x 54, $A_4^2:C_4$ x 30, $C_2^4:C_6^2$ x 21, $C_2^4:C_6^2$ x 18, $A_4^2:C_2^2$ x 16, $C_3\times C_2^3.S_4$ x 6, $(C_2^3\times C_6):C_{12}$ x 6, $C_2^6:C_3^2$ x 4, $C_2^2:S_4\times C_6$ x 3, $C_2^4:(C_6\times S_3)$ x 3, $(C_2^2\times C_6).S_4$ x 2, $C_4\times A_4^2$ x 2, $A_4^2:C_4$ x 2, $A_4^2:C_4$ x 2, $(C_2^3\times C_6):C_{12}$ x 2, $C_3\times C_2^4:C_{12}$ x 2, $(C_2^2\times C_6):S_4$, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $C_6^2.C_2^4$, $C_2^4:C_6^2$

Order 512: $C_2^7:C_4$ x 10, $C_2^5.C_2^4$ x 8, $C_2^4\wr C_2$ x 6, $C_2^6.D_4$ x 4, $C_2\times C_2^6.C_2^2$ x 2, $C_2^8.C_2$ x 2, $C_2^7.C_2^2$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$ x 2, $C_2^7.C_2^2$, $C_2^4.C_2^5$, $C_2^9$

Order 432: $C_3\times A_4^2$ x 69, $A_4\times C_6^2$ x 9, $C_6^2:A_4$ x 6, $C_6^2:C_{12}$ x 2, $C_3\times C_6.S_4$ x 2, $C_6\times S_3\times A_4$, $C_6^2:D_6$

Order 256: $C_2^8$

Order 81: $C_3^4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 384 (order at least 432) are shown, as well as all normal subgroups of any index.

Order 165888: $A_4^2\wr C_2.C_4$

Order 82944: $C_2\times A_4^2\wr C_2$, $C_2.A_4^2\wr C_2$

Order 55296: $C_2^9.C_3:S_3^2$, $C_2^9.(C_3\times S_3^2)$, $C_2^9.(C_3\times S_3^2)$

Order 41472: $C_2\times C_2^8.C_3^4$, $A_4^2\wr C_2$

Order 27648: $C_2^4:A_4^2.C_{12}$ x 2, $C_2^4.A_4^2:D_6$ x 2, $C_2^9.(S_3\times C_3^2)$ x 2, $(C_2^3\times A_4^2).S_4$, $(C_2^3\times A_4^2).S_4$, $C_2^8.(C_3^2\times D_6)$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$

Order 20736: $C_2^8.C_3^4$

Order 18432: $C_2^9.S_3^2$, $C_2^9.(C_6\times S_3)$, $C_2^8.(S_3\times C_{12})$, $C_2^9.S_3^2$

Order 13824: $C_2\times C_2^8.C_3^3$ x 4, $C_2\times C_2^8.C_3^3$ x 4, $C_2\times C_2^8.C_3^3$ x 3, $C_2^4:A_4^2.C_6$ x 2, $C_2^6.C_6^3$, $C_2^8.(S_3\times C_3^2)$, $(C_2^2\times A_4^2).S_4$, $C_2^8:(S_3\times C_3^2)$

Order 10368: $C_2\times C_2^6.C_3^4$

Order 9216: $C_2^9.(C_3\times S_3)$ x 4, $C_2^7.(S_3\times A_4)$ x 2, $C_2^6.(A_4\times D_6)$ x 2, $C_2^7.(C_3\times S_4)$ x 2, $C_2^8.(C_6\times S_3)$ x 2, $C_2^4.C_6^2.C_2^3.C_2$, $C_2\times C_2^4.C_6^2.C_2^3$, $C_2^5.\PSOPlus(4,3)$, $C_2^5.\PSOPlus(4,3)$, $C_2^5.(A_4\times S_4)$, $C_2^5.(A_4\times S_4)$, $C_2^6.(C_6\times S_4)$, $C_2^6.A_4^2$, $C_2^6.(C_{12}\times A_4)$

Order 6912: $C_2^8.C_3^3$ x 4, $C_2^2.A_4^3$ x 4, $C_2^6.C_3^3.C_2^2$ x 3, $C_2^2.A_4^3$ x 3, $A_4^3:C_4$, $A_4^3:C_2^2$, $A_4^3:C_2^2$, $A_4^3:C_4$

Order 6144: $C_2^9.D_6$, $C_2^9.D_6$, $C_2^8.(C_2\times C_{12})$

Order 5184: $C_3\times A_4^3$, $C_6\times A_4^2:S_3$, $(C_3\times A_4^2):C_{12}$

Order 4608: $C_2^9:C_3^2$ x 11, $C_2^9:C_3^2$ x 6, $C_2\times C_2^8.C_3^2$ x 3, $C_2^6.C_6^2.C_2$ x 3, $C_2^6.C_6^2.C_2$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^6.(S_3\times A_4)$ x 2, $(C_2^2\times A_4)\wr C_2$ x 2, $C_2^8:(C_3\times S_3)$ x 2, $C_2\times A_4^2.C_2^4$, $A_4^2.C_2^5$, $C_2\times C_2^6.C_6^2$, $C_2\times C_2^6.C_6^2$, $C_2\times C_2^4.C_6^2.C_2^2$, $C_2\times C_2^4.C_6^2.C_2^2$, $C_2^4:\PSOPlus(4,3)$, $C_2^5:A_4^2$, $C_2^6.(C_3\times S_4)$, $C_2^6:(C_3\times S_4)$, $C_2^9:C_3^2$

Order 3456: $C_2\times A_4^3$ x 24, $C_2\times C_2^6.C_3^3$ x 3, $C_2^4.C_6^3$ x 2, $C_6\times C_2^6.C_3^2$ x 2, $C_6\times C_2^6.C_3^2$, $C_2\times C_2^6.C_3^3$, $A_4^2:S_4$, $A_4^3:C_2$

Order 3072: $C_2^8.C_{12}$ x 2, $C_2^9.C_6$ x 2, $C_2^7.S_4$ x 2, $C_2\times C_2^4.C_2^4:S_3$, $C_2^4:C_3.C_2^5.C_2$, $C_2^6.C_6.C_2^3$, $C_2\times C_2^6.C_6.C_2^2$, $C_2^7.S_4$

Order 2592: $C_6\times C_3\times A_4^2$, $C_3\times C_6\times A_4^2$, $(C_3\times A_4)\wr C_2$

Order 2304: $C_2^4:A_4^2$ x 12, $C_2^4:A_4^2$ x 11, $C_2^4:A_4^2$ x 9, $A_4^2.C_2^4$ x 8, $C_2^4:A_4^2$ x 6, $C_2\times A_4^2:D_4$ x 6, $(C_2^2\times A_4^2):C_4$ x 6, $C_2^8.C_3^2$ x 3, $C_2\times A_4^2.C_2^3$ x 2, $A_4^2.C_2^2.C_2^2$ x 2, $C_2^2\wr C_3:C_{12}$ x 2, $C_2^5.(C_3\times S_4)$ x 2, $C_2^4:(A_4\times D_6)$ x 2, $C_2\times A_4^2.C_2^2.C_2$, $A_4^2:C_2^4$, $(C_2^3\times A_4).S_4$, $(C_2^3\times A_4):S_4$, $(A_4\times C_2^4):C_{12}$, $C_2^4:(C_6\times S_4)$, $A_4\times C_2^3:S_4$, $A_4\times C_2^3.S_4$, $C_2^4:A_4^2$, $A_4\times C_2^4:C_{12}$, $C_2^4:A_4^2$, $C_2^4:(C_6\times S_4)$

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

Order 1728: $A_4^3$ x 22, $C_2^6:C_3^3$ x 6, $C_2^6:C_3^3$ x 4, $C_2^6:C_3^3$ x 3, $A_4^2:D_6$ x 3, $(C_3\times A_4^2):C_4$ x 3, $A_4^2:C_{12}$ x 3, $A_4\wr C_2\times C_6$ x 2, $C_2\times C_6^2:S_4$, $(C_2\times C_6^2).S_4$, $(C_2^3\times C_6^2):C_6$, $(C_2^2\times C_6^2):C_{12}$

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

Order 1296: $C_3^2\times A_4^2$ x 2

Order 1152: $C_2^3:A_4^2$ x 59, $C_2^3\times A_4^2$ x 45, $C_2^3:A_4^2$ x 38, $C_2\times A_4^2:C_4$ x 7, $A_4^2:D_4$ x 4, $C_2^5:C_6^2$ x 4, $A_4^2:C_2^3$ x 3, $C_6\times C_2^6:C_3$ x 3, $C_2^4:(S_3\times A_4)$ x 2, $C_2^4\times C_6\times A_4$ x 2, $(C_2^2\times A_4):S_4$, $C_2^4:(C_3\times S_4)$, $A_4\times C_2^2:S_4$, $A_4\wr C_2\times C_4$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^3:A_4^2$, $(C_2^2\times S_4):A_4$

Order 1024: $C_2^8.C_2^2$, $C_2^9.C_2$, $C_2^8.C_2^2$, $C_2^8.C_2^2$, $C_2^9.C_2$

Order 864: $C_6\times A_4^2$ x 45, $C_2^5:C_3^3$ x 4, $A_4^2:S_3$ x 3, $C_2^2:C_6^3$ x 2, $A_4^2:C_6$ x 2, $C_6^2:S_4$, $(C_6\times D_6):A_4$

Order 768: $C_2^8.C_3$ x 20, $C_2^8.C_3$ x 11, $C_2^6:A_4$ x 9, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^7:C_6$ x 6, $C_2^6:C_{12}$ x 6, $C_2^5.S_4$ x 4, $C_2^6:A_4$ x 4, $C_2^6:A_4$ x 4, $C_2^6:C_{12}$ x 4, $C_2^2\times C_2^2:(A_4:C_4)$ x 3, $C_2^7.C_6$ x 3, $C_2^5:S_4$ x 2, $C_2^5:S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $A_4.(C_2^5.C_2)$, $C_2^7:C_6$, $C_2^6:A_4$, $C_2^7.C_6$

Order 648: $C_6^3:C_3$, $C_3\times C_6.S_3^2$

Order 576: $C_2^2\times A_4^2$ x 80, $C_2^2:A_4^2$ x 51, $C_2^2:A_4^2$ x 32, $A_4^2:C_4$ x 15, $C_2^4:C_6^2$ x 12, $C_2^4:C_6^2$ x 12, $A_4^2:C_2^2$ x 10, $C_2^6:C_3^2$ x 3, $C_3\times C_2^3.S_4$ x 3, $C_2^4:(C_6\times S_3)$ x 3, $(C_2^3\times C_6):C_{12}$ x 3, $C_2^2:S_4\times C_6$ x 2, $(C_2^2\times C_6):S_4$, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $(C_2^2\times C_6).S_4$, $C_4\times A_4^2$, $A_4^2:C_4$, $A_4^2:C_4$, $C_6^2.C_2^4$, $C_2^4:C_6^2$, $(C_2^3\times C_6):C_{12}$, $C_3\times C_2^4:C_{12}$

Order 512: $C_2^7:C_4$ x 6, $C_2^5.C_2^4$ x 4, $C_2^4\wr C_2$ x 3, $C_2^8.C_2$ x 2, $C_2^7.C_2^2$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.D_4$ x 2, $C_2\times C_2^6.C_2^2$, $C_2^7.C_2^2$, $C_2^4.C_2^5$, $C_2^6.C_2^3$, $C_2^9$

Order 432: $C_3\times A_4^2$ x 41, $A_4\times C_6^2$ x 6, $C_6^2:A_4$ x 4, $C_6\times S_3\times A_4$, $C_6^2:D_6$, $C_6^2:C_{12}$, $C_3\times C_6.S_4$

Order 256: $C_2^8$

Order 81: $C_3^4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 165888: $A_4^2\wr C_2.C_4$ ($C_1$)

Order 82944: $C_2.A_4^2\wr C_2$ ($C_2$) x 2, $C_2\times A_4^2\wr C_2$ ($C_2$)

Order 55296: $C_2^9.C_3:S_3^2$ ($C_3$)

Order 41472: $A_4^2\wr C_2$ ($C_4$) x 2, $C_2\times C_2^8.C_3^4$ ($C_2^2$)

Order 27648: $C_2^4:A_4^2.C_{12}$ ($C_6$) x 2, $C_2^4:A_4^2.C_{12}$ ($S_3$) x 2, $C_2^4.A_4^2:D_6$ ($C_6$), $C_2^4.A_4^2:D_6$ ($S_3$)

Order 20736: $C_2^8.C_3^4$ ($C_2\times C_4$)

Order 13824: $C_2\times C_2^8.C_3^3$ ($D_6$) x 3, $C_2^4:A_4^2.C_6$ ($C_{12}$) x 2, $C_2^4:A_4^2.C_6$ ($C_3:C_4$) x 2, $C_2\times C_2^8.C_3^3$ ($C_2\times C_6$)

Order 9216: $C_2^5.\PSOPlus(4,3)$ ($C_3\times S_3$) x 2, $C_2^5.\PSOPlus(4,3)$ ($C_3\times S_3$)

Order 6912: $C_2^2.A_4^3$ ($C_4\times S_3$) x 2, $C_2^2.A_4^3$ ($C_2\times C_{12}$), $C_2^2.A_4^3$ ($C_6:C_4$)

Order 4608: $C_2^9:C_3^2$ ($C_6\times S_3$) x 3, $C_2^9:C_3^2$ ($S_3^2$) x 3, $C_2^4:\PSOPlus(4,3)$ ($C_3:C_{12}$) x 2

Order 2304: $C_2^4:A_4^2$ ($S_3\times C_{12}$) x 2, $C_2^4:A_4^2$ ($C_6.D_6$) x 2, $C_2^4:A_4^2$ ($C_6:C_{12}$), $C_2^4:A_4^2$ ($C_6.D_6$)

Order 1536: $C_2^9.C_3$ ($C_3\times S_3^2$) x 3, $C_2^9.C_3$ ($C_3:S_3^2$)

Order 768: $C_2^6:A_4$ ($C_6.S_3^2$) x 2, $C_2^6:A_4$ ($C_6.S_3^2$), $C_2^6:A_4$ ($C_6.S_3^2$)

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

Order 256: $C_2^8$ ($C_3\times C_6.S_3^2$)

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

Order 1: $C_1$ ($A_4^2\wr C_2.C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 165888: $A_4^2\wr C_2.C_4$ ($C_1$)

Order 82944: $C_2\times A_4^2\wr C_2$ ($C_2$), $C_2.A_4^2\wr C_2$ ($C_2$)

Order 55296: $C_2^9.C_3:S_3^2$ ($C_3$)

Order 41472: $C_2\times C_2^8.C_3^4$ ($C_2^2$), $A_4^2\wr C_2$ ($C_4$)

Order 27648: $C_2^4:A_4^2.C_{12}$ ($C_6$), $C_2^4:A_4^2.C_{12}$ ($S_3$), $C_2^4.A_4^2:D_6$ ($C_6$), $C_2^4.A_4^2:D_6$ ($S_3$)

Order 20736: $C_2^8.C_3^4$ ($C_2\times C_4$)

Order 13824: $C_2\times C_2^8.C_3^3$ ($D_6$) x 2, $C_2\times C_2^8.C_3^3$ ($C_2\times C_6$), $C_2^4:A_4^2.C_6$ ($C_{12}$), $C_2^4:A_4^2.C_6$ ($C_3:C_4$)

Order 9216: $C_2^5.\PSOPlus(4,3)$ ($C_3\times S_3$), $C_2^5.\PSOPlus(4,3)$ ($C_3\times S_3$)

Order 6912: $C_2^2.A_4^3$ ($C_2\times C_{12}$), $C_2^2.A_4^3$ ($C_6:C_4$), $C_2^2.A_4^3$ ($C_4\times S_3$)

Order 4608: $C_2^9:C_3^2$ ($C_6\times S_3$) x 2, $C_2^9:C_3^2$ ($S_3^2$) x 2, $C_2^4:\PSOPlus(4,3)$ ($C_3:C_{12}$)

Order 2304: $C_2^4:A_4^2$ ($C_6:C_{12}$), $C_2^4:A_4^2$ ($S_3\times C_{12}$), $C_2^4:A_4^2$ ($C_6.D_6$), $C_2^4:A_4^2$ ($C_6.D_6$)

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

Order 768: $C_2^6:A_4$ ($C_6.S_3^2$), $C_2^6:A_4$ ($C_6.S_3^2$), $C_2^6:A_4$ ($C_6.S_3^2$)

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

Order 256: $C_2^8$ ($C_3\times C_6.S_3^2$)

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

Order 1: $C_1$ ($A_4^2\wr C_2.C_4$)

Series

Derived series

$A_4^2\wr C_2.C_4$

$\rhd$

$C_2^2.A_4^3$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$A_4^2\wr C_2.C_4$

$\rhd$

$C_2.A_4^2\wr C_2$

$\rhd$

$C_2\times C_2^8.C_3^4$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^2.A_4^3$

$\rhd$

$C_2^4:A_4^2$

$\rhd$

$C_2^6:A_4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Lower central series

$A_4^2\wr C_2.C_4$

$\rhd$

$C_2^2.A_4^3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 11 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

The $212 \times 212$ character table is not available for this group.

Rational character table

The $120 \times 120$ rational character table is not available for this group.