Abstract group 27648.bf: $C_2^8.(C_3^2\times D_6)$

• Label: 27648.bf
• Order: \( 2^{10} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $18$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_2^8.(C_3^2\times D_6)$
Order:	\(27648\)\(\medspace = 2^{10} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$\GL(2,4):C_2^5$, of order \(442368\)\(\medspace = 2^{14} \cdot 3^{3} \)
Composition factors:	$C_2$ x 10, $C_3$ 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	607	2672	1440	18320	4608	27648
Conjugacy classes	1	29	17	6	115	8	176
Divisions	1	29	9	6	59	4	108
Autjugacy classes	1	18	7	3	34	2	65

Dimension	1	2	3	4	6	9	12	18	27	36	54
Irr. complex chars.	36	18	12	0	60	12	0	30	4	0	4	176
Irr. rational chars.	4	18	4	8	12	4	26	14	4	10	4	108

Minimal presentations

Permutation degree:	$18$
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:	${\langle a, b, c, d, e, f, g, h, i, j \mid a^{6}=b^{6}=c^{6}=d^{2}=e^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(2,3,5)(4,6,8)(10,12,15)(11,13,14)(17,18), (1,2)(3,4,5,6,7,8)(9,10,11,12,14,15)(13,16)\rangle$
Transitive group:	36T13061				more information
Direct product:	$C_2$ $\, \times\, $ $(C_2^8:(S_3\times C_3^2))$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^9:C_3^2)$ . $S_3$	$C_2^9$ . $(S_3\times C_3^2)$	$(A_4^2:C_2^4)$ . $A_4$	$A_4^2$ . $(C_2^4:A_4)$	all 54

Elements of the group are displayed as permutations of degree 18.

Homology

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

Subgroups

There are 9168680 subgroups in 173480 conjugacy classes, 72 normal (46 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^8:(S_3\times C_3^2)$
Commutator:	$G' \simeq$ $C_2^8.C_3$	$G/G' \simeq$ $C_6^2$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $A_4^3:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9$	$G/\operatorname{Fit} \simeq$ $S_3\times C_3^2$
Radical:	$R \simeq$ $C_2^8.(C_3^2\times D_6)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^7$	$G/\operatorname{soc} \simeq$ $C_6^2:C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2$
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 144 (order at least 192) are shown, as well as all normal subgroups of any index.

Order 27648: $C_2^8.(C_3^2\times D_6)$

Order 13824: $C_2^8:(S_3\times C_3^2)$ x 2, $C_2\times C_2^4:C_3\times A_4^2$

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

Order 6912: $C_2^4:C_3\times A_4^2$, $A_4^3:C_2^2$

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

Order 3456: $C_2\times A_4^3$ x 3, $A_4^3:C_2$ x 2, $C_2\times C_2^4:C_3\times C_3\times A_4$

Order 3072: $C_2\times C_2^6.C_6.C_2^2$ x 2, $C_2^9.C_6$ x 2, $C_2\times C_2^4.C_2^4:S_3$

Order 2304: $C_2\times A_4^2:D_4$ x 8, $C_2^4\times A_4^2$ x 7, $(C_2^2\times A_4^2):C_4$ x 4, $C_2^4:A_4^2$ x 3, $C_2^4:A_4^2$ x 3, $C_2^8.C_3^2$ x 2, $C_2^4:A_4^2$ x 2, $C_2^4:A_4^2$ x 2, $C_2^4:(A_4\times D_6)$ x 2, $A_4^2.C_2^3.C_2$, $A_4^2:C_2^4$, $C_2^4:(C_6\times S_4)$, $A_4\times C_2^3:S_4$

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

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

Order 1152: $C_2^3:A_4^2$ x 39, $C_2^3\times A_4^2$ x 36, $C_2^3:A_4^2$ x 11, $A_4^2:D_4$ x 8, $C_2^4:(S_3\times A_4)$ x 4, $A_4^2:C_2^3$ x 4, $C_2\times A_4^2:C_4$ x 4, $C_2^3:A_4^2$ x 4, $A_4\times C_2^2:S_4$ x 2, $(C_2^2\times S_4):A_4$ x 2, $C_6\times C_2^6:C_3$ x 2, $C_2^4\times C_6\times A_4$, $C_2^5:C_6^2$

Order 1024: $C_2^9.C_2$

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

Order 768: $C_2^8.C_3$ x 23, $C_2^8.C_3$ x 19, $C_2^7:C_6$ x 16, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^6:C_{12}$ x 8, $C_2^6:A_4$ x 6, $C_2^6:A_4$ x 6, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^6:A_4$ x 3, $C_2^7:C_6$ x 2, $C_2^6:A_4$ x 2, $C_2^7.C_6$ x 2, $C_2^5:S_4$, $C_2^2\times C_2^2:(A_4:C_4)$, $C_2^3:\GL(2,\mathbb{Z}/4)$

Order 576: $C_2^2\times A_4^2$ x 47, $C_2^2:A_4^2$ x 37, $C_2^4:C_6^2$ x 11, $C_2^2:A_4^2$ x 11, $A_4^2:C_2^2$ x 6, $C_2^4:C_6^2$ x 3, $C_2^4:(C_6\times S_3)$ x 3, $C_2^6:C_3^2$ x 2, $A_4^2:C_4$ x 2, $C_2^4:C_6^2$ x 2, $C_2^2:S_4\times C_6$, $A_4^2:C_2^2$, $C_6^2.C_2^4$

Order 512: $C_2^4\wr C_2$ x 8, $C_2^8.C_2$ x 3, $C_2^7:C_4$ x 3, $C_2^9$

Order 432: $C_3\times A_4^2$ x 6, $C_6^2:A_4$, $C_6\times S_3\times A_4$

Order 384: $A_4\times C_2^5$ x 350, $C_2^5:A_4$ x 99, $C_2^5:A_4$ x 55, $C_2\wr C_6$ x 16, $C_2^5:A_4$ x 12, $C_2^5:C_{12}$ x 8, $C_2\wr S_3$ x 8, $C_2^6:C_6$ x 8, $C_2^5:A_4$ x 8, $C_2^2\wr S_3$ x 4, $C_2^4.S_4$ x 4, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^6:C_6$ x 2, $C_2^5:C_{12}$ x 2, $C_2^6\times C_6$, $C_2^2\times \GL(2,\mathbb{Z}/4)$, $C_2^2.\GL(2,\mathbb{Z}/4)$

Order 288: $C_2\times A_4^2$ x 85, $C_2^3:C_6^2$ x 61, $C_2^5:C_3^2$ x 32, $(C_2\times D_6):A_4$ x 6, $A_4\wr C_2$ x 6, $C_6^2:D_4$ x 4, $C_3\times C_2^3:A_4$ x 4, $C_3\times C_2^2:S_4$ x 2, $A_4\times S_4$ x 2, $C_6^2.D_4$, $C_2^3\times C_6^2$, $C_6^2:D_4$

Order 256: $C_2^5:D_4$ x 64, $C_2^6:C_4$ x 32, $C_2^8$ x 27, $C_2^6:C_4$ x 13, $C_2^5:D_4$ x 7

Order 216: $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 2

Order 192: $A_4\times C_2^4$ x 1428, $C_2^2\wr C_3$ x 145, $C_2^4:A_4$ x 51, $C_2^4:A_4$ x 24, $C_2^5\times C_6$ x 23, $C_2^5:C_6$ x 16, $C_2^4:C_{12}$ x 8, $C_2\times \GL(2,\mathbb{Z}/4)$ x 8, $C_2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:C_{12}$ x 4, $C_2^3:S_4$ x 4, $C_2^4:C_{12}$ x 2, $C_2^3.S_4$ x 2, $C_2^5:C_6$ x 2, $C_2^3\times S_4$, $C_2^3.S_4$, $C_2^4:D_6$

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 144 (order at least 192) are shown, as well as all normal subgroups of any index.

Order 27648: $C_2^8.(C_3^2\times D_6)$

Order 13824: $C_2\times C_2^4:C_3\times A_4^2$, $C_2^8:(S_3\times C_3^2)$

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

Order 6912: $C_2^4:C_3\times A_4^2$, $A_4^3:C_2^2$

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

Order 3456: $C_2\times A_4^3$ x 2, $C_2\times C_2^4:C_3\times C_3\times A_4$, $A_4^3:C_2$

Order 3072: $C_2\times C_2^6.C_6.C_2^2$ x 2, $C_2\times C_2^4.C_2^4:S_3$, $C_2^9.C_6$

Order 2304: $C_2^4\times A_4^2$ x 5, $C_2\times A_4^2:D_4$ x 4, $C_2^4:A_4^2$ x 3, $C_2^4:A_4^2$ x 2, $C_2^4:A_4^2$ x 2, $C_2^4:A_4^2$ x 2, $(C_2^2\times A_4^2):C_4$ x 2, $A_4^2.C_2^3.C_2$, $C_2^8.C_3^2$, $A_4^2:C_2^4$, $C_2^4:(C_6\times S_4)$, $A_4\times C_2^3:S_4$, $C_2^4:(A_4\times D_6)$

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

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

Order 1152: $C_2^3:A_4^2$ x 18, $C_2^3\times A_4^2$ x 17, $C_2^3:A_4^2$ x 4, $A_4^2:C_2^3$ x 3, $C_2\times A_4^2:C_4$ x 3, $A_4^2:D_4$ x 2, $C_2^3:A_4^2$ x 2, $C_2^4:(S_3\times A_4)$, $A_4\times C_2^2:S_4$, $(C_2^2\times S_4):A_4$, $C_2^4\times C_6\times A_4$, $C_2^5:C_6^2$, $C_6\times C_2^6:C_3$

Order 1024: $C_2^9.C_2$

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

Order 768: $C_2^8.C_3$ x 15, $C_2^8.C_3$ x 11, $C_2^7:C_6$ x 10, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^6:C_{12}$ x 4, $C_2^6:A_4$ x 3, $C_2^6:A_4$ x 3, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:A_4$ x 2, $C_2^7:C_6$ x 2, $C_2^6:A_4$ x 2, $C_2^7.C_6$ x 2, $C_2^5:S_4$, $C_2^2\times C_2^2:(A_4:C_4)$, $C_2^3:\GL(2,\mathbb{Z}/4)$

Order 576: $C_2^2\times A_4^2$ x 22, $C_2^2:A_4^2$ x 16, $C_2^4:C_6^2$ x 5, $C_2^2:A_4^2$ x 4, $A_4^2:C_2^2$ x 4, $C_2^4:C_6^2$ x 3, $C_2^4:C_6^2$ x 2, $C_2^4:(C_6\times S_3)$ x 2, $C_2^6:C_3^2$, $C_2^2:S_4\times C_6$, $A_4^2:C_2^2$, $A_4^2:C_4$, $C_6^2.C_2^4$

Order 512: $C_2^4\wr C_2$ x 4, $C_2^8.C_2$ x 3, $C_2^7:C_4$ x 3, $C_2^9$

Order 432: $C_3\times A_4^2$ x 3, $C_6^2:A_4$, $C_6\times S_3\times A_4$

Order 384: $A_4\times C_2^5$ x 80, $C_2^5:A_4$ x 49, $C_2^5:A_4$ x 19, $C_2\wr C_6$ x 6, $C_2^5:C_{12}$ x 6, $C_2^5:A_4$ x 6, $C_2^6:C_6$ x 4, $C_2^2\wr S_3$ x 3, $C_2^4.S_4$ x 3, $C_2^5:A_4$ x 3, $C_2^6:C_6$ x 2, $C_2\wr S_3$ x 2, $C_2^5:C_{12}$ x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6\times C_6$, $C_2^2\times \GL(2,\mathbb{Z}/4)$, $C_2^2.\GL(2,\mathbb{Z}/4)$

Order 288: $C_2\times A_4^2$ x 31, $C_2^3:C_6^2$ x 18, $C_2^5:C_3^2$ x 14, $C_6^2:D_4$ x 2, $C_3\times C_2^3:A_4$ x 2, $(C_2\times D_6):A_4$ x 2, $A_4\wr C_2$ x 2, $C_6^2.D_4$, $C_2^3\times C_6^2$, $C_3\times C_2^2:S_4$, $A_4\times S_4$, $C_6^2:D_4$

Order 256: $C_2^5:D_4$ x 29, $C_2^8$ x 17, $C_2^6:C_4$ x 13, $C_2^6:C_4$ x 10, $C_2^5:D_4$ x 6

Order 216: $C_6^2:C_6$ x 3, $C_6^2:C_6$

Order 192: $A_4\times C_2^4$ x 240, $C_2^2\wr C_3$ x 60, $C_2^4:A_4$ x 17, $C_2^5\times C_6$ x 11, $C_2^4:A_4$ x 11, $C_2^5:C_6$ x 10, $C_2^4:C_{12}$ x 4, $C_2\times \GL(2,\mathbb{Z}/4)$ x 4, $C_2^3:S_4$ x 3, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:C_{12}$ x 2, $C_2^4:C_{12}$ x 2, $C_2^5:C_6$ x 2, $C_2^3\times S_4$, $C_2^3.S_4$, $C_2^3.S_4$, $C_2^4:D_6$

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 27648: $C_2^8.(C_3^2\times D_6)$ ($C_1$)

Order 13824: $C_2^8:(S_3\times C_3^2)$ ($C_2$) x 2, $C_2\times C_2^4:C_3\times A_4^2$ ($C_2$)

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

Order 6912: $C_2^4:C_3\times A_4^2$ ($C_2^2$)

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

Order 3072: $C_2\times C_2^4.C_2^4:S_3$ ($C_3^2$)

Order 2304: $C_2^4:A_4^2$ ($C_2\times C_6$) x 2, $C_2^4\times A_4^2$ ($C_2\times C_6$), $A_4^2:C_2^4$ ($A_4$), $C_2^4:A_4^2$ ($C_2\times C_6$), $C_2^4:A_4^2$ ($D_6$)

Order 1536: $C_2^4:\GL(2,\mathbb{Z}/4)$ ($C_3\times C_6$) x 2, $C_2^9.C_3$ ($C_3\times S_3$) x 2, $C_2^9.C_3$ ($C_3\times S_3$) x 2, $C_2^9.C_3$ ($C_3\times C_6$)

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

Order 768: $C_2^8.C_3$ ($C_6\times S_3$) x 2, $C_2^6:A_4$ ($C_6\times S_3$) x 2, $C_2^5:S_4$ ($C_3\times A_4$), $C_2^8.C_3$ ($C_6^2$)

Order 576: $C_2^2\times A_4^2$ ($C_2^2\times A_4$)

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

Order 384: $C_2^2\wr S_3$ ($C_6\times A_4$) x 2, $C_2^5:A_4$ ($C_6\times A_4$), $C_2^5:A_4$ ($S_3\times A_4$)

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

Order 256: $C_2^8$ ($C_3^2\times D_6$)

Order 192: $C_2^2\wr C_3$ ($C_2^2:C_6^2$), $C_2^2\wr C_3$ ($A_4\times D_6$)

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

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

Order 96: $C_2^3:A_4$ ($C_3\times C_2^3:A_4$), $C_2^3:A_4$ ($(C_2\times D_6):A_4$), $C_2^3:A_4$ ($A_4\wr C_2$)

Order 64: $C_2^6$ ($C_6\times S_3\times A_4$)

Order 48: $C_2^2:A_4$ ($A_4^2:C_2^2$), $C_2^2:A_4$ ($C_2^4:C_6^2$), $C_2^2:A_4$ ($C_2^4:(C_6\times S_3)$)

Order 32: $C_2^5$ ($A_4^2:C_6$), $C_2^5$ ($(C_6\times D_6):A_4$)

Order 16: $C_2^4$ ($A_4\wr C_2\times C_6$), $C_2^4$ ($(C_2^3\times C_6^2):C_6$)

Order 8: $C_2^3$ ($A_4^3:C_2$)

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

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

Order 1: $C_1$ ($C_2^8.(C_3^2\times D_6)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 27648: $C_2^8.(C_3^2\times D_6)$ ($C_1$)

Order 13824: $C_2\times C_2^4:C_3\times A_4^2$ ($C_2$), $C_2^8:(S_3\times C_3^2)$ ($C_2$)

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

Order 6912: $C_2^4:C_3\times A_4^2$ ($C_2^2$)

Order 4608: $C_2^5\times A_4^2$ ($C_6$), $C_2^9:C_3^2$ ($C_6$), $C_2^9:C_3^2$ ($S_3$), $C_2^9:C_3^2$ ($C_6$), $C_2^6:(C_3\times S_4)$ ($C_6$), $C_2^6.(S_3\times A_4)$ ($C_6$), $(C_2^2\times A_4)\wr C_2$ ($C_6$)

Order 3072: $C_2\times C_2^4.C_2^4:S_3$ ($C_3^2$)

Order 2304: $C_2^4\times A_4^2$ ($C_2\times C_6$), $A_4^2:C_2^4$ ($A_4$), $C_2^4:A_4^2$ ($C_2\times C_6$), $C_2^4:A_4^2$ ($D_6$), $C_2^4:A_4^2$ ($C_2\times C_6$)

Order 1536: $C_2^9.C_3$ ($C_3\times S_3$) x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ ($C_3\times C_6$), $C_2^9.C_3$ ($C_3\times C_6$), $C_2^9.C_3$ ($C_3\times S_3$)

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

Order 768: $C_2^8.C_3$ ($C_6\times S_3$) x 2, $C_2^5:S_4$ ($C_3\times A_4$), $C_2^8.C_3$ ($C_6^2$), $C_2^6:A_4$ ($C_6\times S_3$)

Order 576: $C_2^2\times A_4^2$ ($C_2^2\times A_4$)

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

Order 384: $C_2^5:A_4$ ($C_6\times A_4$), $C_2^5:A_4$ ($S_3\times A_4$), $C_2^2\wr S_3$ ($C_6\times A_4$)

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

Order 256: $C_2^8$ ($C_3^2\times D_6$)

Order 192: $C_2^2\wr C_3$ ($C_2^2:C_6^2$), $C_2^2\wr C_3$ ($A_4\times D_6$)

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

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

Order 96: $C_2^3:A_4$ ($C_3\times C_2^3:A_4$), $C_2^3:A_4$ ($(C_2\times D_6):A_4$), $C_2^3:A_4$ ($A_4\wr C_2$)

Order 64: $C_2^6$ ($C_6\times S_3\times A_4$)

Order 48: $C_2^2:A_4$ ($A_4^2:C_2^2$), $C_2^2:A_4$ ($C_2^4:C_6^2$), $C_2^2:A_4$ ($C_2^4:(C_6\times S_3)$)

Order 32: $C_2^5$ ($A_4^2:C_6$), $C_2^5$ ($(C_6\times D_6):A_4$)

Order 16: $C_2^4$ ($A_4\wr C_2\times C_6$), $C_2^4$ ($(C_2^3\times C_6^2):C_6$)

Order 8: $C_2^3$ ($A_4^3:C_2$)

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

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

Order 1: $C_1$ ($C_2^8.(C_3^2\times D_6)$)

Series

Derived series

$C_2^8.(C_3^2\times D_6)$

$\rhd$

$C_2^8.C_3$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2^8.(C_3^2\times D_6)$

$\rhd$

$C_2\times C_2^4:C_3\times A_4^2$

$\rhd$

$C_2^4:C_3\times A_4^2$

$\rhd$

$C_2^4\times A_4^2$

$\rhd$

$C_2^8.C_3$

$\rhd$

$C_2^2\wr C_3$

$\rhd$

$C_2^2:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$C_2^8.(C_3^2\times D_6)$

$\rhd$

$C_2^8.C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $176 \times 176$ 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 $108 \times 108$ rational character table.