Abstract group 1769472.em: $C_2\times C_2^{10}.C_6^2:D_{12}$

• Label: 1769472.em
• Order: \( 2^{16} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{19} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: not computed
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_2\times C_2^{10}.C_6^2:D_{12}$
Order:	\(1769472\)\(\medspace = 2^{16} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2\times C_2^8.C_6^2.A_4.C_2^5.C_2$, of order \(14155776\)\(\medspace = 2^{19} \cdot 3^{3} \)
Composition factors:	$C_2$ x 16, $C_3$ x 3
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	17407	19520	396288	433088	110592	792576	1769472
Conjugacy classes	1	113	5	142	75	8	40	384
Divisions	1	113	5	142	63	6	30	360
Autjugacy classes	1	89	5	72	49	3	11	230

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n, o \mid b^{12}=c^{6}=d^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,32,28,9,13,23,2,31,27,10,14,24)(3,36,29,8,15,21)(4,35,30,7,16,22)(5,33,25,11,18,19,6,34,26,12,17,20) \!\cdots\! \rangle$
Transitive group:	36T42302	36T42303			more information
Direct product:	$C_2$ $\, \times\, $ $(C_2^{10}.C_6^2:D_{12})$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^{13}$ . $(C_3^2:D_{12})$	$C_2^{11}$ . $(C_6^2:D_{12})$	$(C_2^8.A_4^2:C_4)$ . $D_6$	$(C_2^{12}.C_6:S_3^2)$ . $C_2$	all 41

Elements of the group are displayed as permutations of degree 36.

Homology

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

Subgroups

There are 54 normal subgroups (40 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^{10}.C_6^2:D_{12}$
Commutator:	$G' \simeq$ $C_2^8.C_6^2.A_4.C_2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^9.C_6^2:D_{12}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^{13}$	$G/\operatorname{Fit} \simeq$ $C_3^2:D_{12}$
Radical:	$R \simeq$ $C_2\times C_2^{10}.C_6^2:D_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^{11}$	$G/\operatorname{soc} \simeq$ $C_6^2:D_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^{10}.C_2^5.C_2$
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 18 (order at least 98304) are shown, as well as all normal subgroups of any index.

Order 1769472: $C_2\times C_2^{10}.C_6^2:D_{12}$

Order 884736: $C_2^{10}.C_6^2:D_{12}$ x 4, $C_2^{12}.C_3^3.C_2^3$, $C_2^{12}.C_6:S_3^2$, $C_2\times C_2^{10}.C_6^2:C_{12}$

Order 589824: $C_2^8.C_6^2.C_2^4.C_2^2$

Order 442368: $C_2^{12}.C_3^2.D_6$ x 3, $C_2^{10}.A_4:S_3^2$ x 3, $C_2^{10}.C_6^2:C_{12}$ x 2, $C_2^{12}.C_3^3.C_2^2$, $C_2^8.C_6^2.A_4.C_2^2$, $C_2^9.C_6^2:D_{12}$, $C_2^7.A_4^2:S_4$

Order 294912: $C_2^{12}.\SOPlus(4,2)$ x 8, $C_2^8.C_3^2.C_2^5.C_2^2$, $C_2^8.C_3^2.C_2^5.C_2^2$, $C_2^8.C_6^2.C_2^4.C_2$, $C_2^8.C_6^2.C_2^5$, $C_2^8.C_6^2.C_2^5$, $C_2^8.C_3^2.C_2^5.C_2^2$, $C_2^{12}.C_3^2.C_2^3$, $C_2^8.C_3^2.C_2^5.C_2^2$, $C_2\times C_2^{12}.S_3^2$

Order 221184: $C_2^8.C_6^2:D_{12}$ x 4, $C_2^{12}.C_3^3.C_2$ x 2, $C_2^8.C_6^2.A_4.C_2$ x 2, $C_2^{10}.C_3^3.C_2^3$ x 2, $C_2^9.A_4:S_3^2$ x 2, $C_2^8.C_6^2:S_4$ x 2, $C_2^{12}.C_3^3.C_2$, $C_2^9.C_6^2:C_{12}$

Order 196608: $C_2^{12}.C_6.C_2^3$

Order 147456: $C_2^8.C_6^2.C_2^4$ x 10, $C_2^7.S_4\wr C_2$ x 10, $C_2^{10}.D_6^2$ x 10, $C_2^7.S_4\wr C_2$ x 6, $C_2^7.S_4\wr C_2$ x 6, $C_2^{12}.S_3^2$ x 4, $C_2^7.S_4\wr C_2$ x 4, $C_2^{12}.S_3^2$ x 4, $C_2^8.A_4^2:C_4$ x 4, $C_2^8.C_6^2.C_2^4$ x 3, $C_2^8.C_6^2.C_2^4$ x 3, $C_2^8.C_6^2.C_2^4$ x 2, $C_2^8.C_6^2.C_2^4$ x 2, $C_2^8.C_3^2.C_2^4.C_2^2$ x 2, $C_2^8.C_6^2.C_2^4$ x 2, $C_2^8.C_6^2.C_2^4$ x 2, $C_2^8.(C_2\times D_6:D_6).C_2$ x 2, $C_2^8.C_3^2.C_2^5.C_2$ x 2, $C_2^8.(C_2\times D_6:D_6).C_2$ x 2, $C_2^4:C_3.C_2^6.A_4.C_2^2$, $C_2^4:C_3.C_2^6.C_6.C_2^3$, $C_2^8.(C_2\times D_6:D_6).C_2$, $C_2^{12}.(C_6\times S_3)$, $C_2^8.C_3^2.C_2^4.C_2^2$, $C_2^8.(C_2\times D_6:D_6).C_2$, $C_2^8.C_6^2.C_2^4$, $C_2^{12}.C_3.D_6$, $C_2^8.C_3^2.C_2^5.C_2$, $C_2^{10}.C_6^2.C_2^2$, $C_2^8.C_6^2.C_2^4$, $C_2^{10}.(A_4\times D_6)$, $C_2^9.\PSOPlus(4,3)$, $C_2^{10}.(C_6\times S_4)$, $C_2^9.(A_4\times S_4)$

Order 110592: $C_2^6.A_4^2:D_6$ x 7, $C_2^8.A_4:S_3^2$ x 7, $C_2^8.C_6^2.C_6.C_2$ x 3, $C_2^{10}.C_3^3.C_2^2$ x 3, $C_2^5.A_4^2:S_4$ x 3, $C_2^8.C_6^2:C_{12}$ x 2, $C_2^{10}.C_3^3.C_2^2$, $C_2^{12}.C_3^3$, $C_2^9.C_3^2:D_{12}$

Order 98304: $C_2^{12}.D_{12}$ x 4, $C_2^{12}.C_6.C_2^2$ x 2, $C_2^4:C_3.C_2^6.C_2^5$, $C_2^{12}.C_6.C_2^2$, $C_2^8.C_6.C_2^5.C_2$

Order 73728: $C_2^8.C_6^2.C_2^3$ x 2, $C_2^8.C_3:S_3.C_2^4$, $C_2^8.C_6^2.C_2^3$

Order 65536: $C_2^{10}.C_2^5.C_2$

Order 36864: $C_2^6.A_4^2:C_4$ x 2, $C_2^8.C_6^2.C_2^2$, $C_2^8.C_6^2.C_2^2$

Order 24576: $C_2^{12}.C_6$

Order 18432: $C_2^6.\PSOPlus(4,3)$ x 2, $C_2^8.C_6^2.C_2$

Order 12288: $C_2^{12}.C_3$

Order 9216: $C_2^{10}:C_3^2$, $C_2^5.\PSOPlus(4,3)$

Order 8192: $C_2^{13}$

Order 6912: $(C_2^3\times C_6^2):D_{12}$

Order 4608: $C_2^4:\PSOPlus(4,3)$ x 2, $C_2^9:C_3^2$

Order 4096: $C_2^{12}$

Order 2304: $C_2^4:A_4^2$

Order 2048: $C_2^{11}$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 32: $C_2^5$

Order 27: $C_3^3$

Order 16: $C_2^4$

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 18 (order at least 98304) are shown, as well as all normal subgroups of any index.

Order 1769472: $C_2\times C_2^{10}.C_6^2:D_{12}$

Order 884736: $C_2^{12}.C_3^3.C_2^3$, $C_2^{12}.C_6:S_3^2$, $C_2\times C_2^{10}.C_6^2:C_{12}$, $C_2^{10}.C_6^2:D_{12}$

Order 589824: $C_2^8.C_6^2.C_2^4.C_2^2$

Order 442368: $C_2^{12}.C_3^2.D_6$ x 2, $C_2^{10}.A_4:S_3^2$ x 2, $C_2^{12}.C_3^3.C_2^2$, $C_2^8.C_6^2.A_4.C_2^2$, $C_2^9.C_6^2:D_{12}$, $C_2^7.A_4^2:S_4$, $C_2^{10}.C_6^2:C_{12}$

Order 294912: $C_2^{12}.\SOPlus(4,2)$ x 2, $C_2^8.C_3^2.C_2^5.C_2^2$, $C_2^8.C_3^2.C_2^5.C_2^2$, $C_2^8.C_6^2.C_2^4.C_2$, $C_2^8.C_6^2.C_2^5$, $C_2^8.C_6^2.C_2^5$, $C_2^8.C_3^2.C_2^5.C_2^2$, $C_2^{12}.C_3^2.C_2^3$, $C_2^8.C_3^2.C_2^5.C_2^2$, $C_2\times C_2^{12}.S_3^2$

Order 221184: $C_2^8.C_6^2.A_4.C_2$ x 2, $C_2^{10}.C_3^3.C_2^3$ x 2, $C_2^9.A_4:S_3^2$ x 2, $C_2^{12}.C_3^3.C_2$, $C_2^{12}.C_3^3.C_2$, $C_2^9.C_6^2:C_{12}$, $C_2^8.C_6^2:D_{12}$, $C_2^8.C_6^2:S_4$

Order 196608: $C_2^{12}.C_6.C_2^3$

Order 147456: $C_2^8.C_6^2.C_2^4$ x 6, $C_2^{10}.D_6^2$ x 6, $C_2^7.S_4\wr C_2$ x 4, $C_2^8.C_6^2.C_2^4$ x 2, $C_2^{12}.S_3^2$ x 2, $C_2^8.C_6^2.C_2^4$ x 2, $C_2^8.C_6^2.C_2^4$ x 2, $C_2^8.C_6^2.C_2^4$ x 2, $C_2^8.(C_2\times D_6:D_6).C_2$ x 2, $C_2^8.C_6^2.C_2^4$ x 2, $C_2^8.C_6^2.C_2^4$ x 2, $C_2^8.C_3^2.C_2^5.C_2$ x 2, $C_2^7.S_4\wr C_2$ x 2, $C_2^7.S_4\wr C_2$ x 2, $C_2^{12}.S_3^2$ x 2, $C_2^8.A_4^2:C_4$ x 2, $C_2^4:C_3.C_2^6.A_4.C_2^2$, $C_2^4:C_3.C_2^6.C_6.C_2^3$, $C_2^8.C_3^2.C_2^4.C_2^2$, $C_2^8.(C_2\times D_6:D_6).C_2$, $C_2^{12}.(C_6\times S_3)$, $C_2^8.C_3^2.C_2^4.C_2^2$, $C_2^8.(C_2\times D_6:D_6).C_2$, $C_2^8.C_6^2.C_2^4$, $C_2^{12}.C_3.D_6$, $C_2^8.C_3^2.C_2^5.C_2$, $C_2^{10}.C_6^2.C_2^2$, $C_2^8.(C_2\times D_6:D_6).C_2$, $C_2^8.C_6^2.C_2^4$, $C_2^{10}.(A_4\times D_6)$, $C_2^7.S_4\wr C_2$, $C_2^9.\PSOPlus(4,3)$, $C_2^{10}.(C_6\times S_4)$, $C_2^9.(A_4\times S_4)$

Order 110592: $C_2^6.A_4^2:D_6$ x 4, $C_2^8.A_4:S_3^2$ x 4, $C_2^8.C_6^2.C_6.C_2$ x 3, $C_2^{10}.C_3^3.C_2^2$ x 2, $C_2^5.A_4^2:S_4$ x 2, $C_2^{10}.C_3^3.C_2^2$, $C_2^{12}.C_3^3$, $C_2^8.C_6^2:C_{12}$, $C_2^9.C_3^2:D_{12}$

Order 98304: $C_2^{12}.C_6.C_2^2$ x 2, $C_2^4:C_3.C_2^6.C_2^5$, $C_2^{12}.C_6.C_2^2$, $C_2^8.C_6.C_2^5.C_2$, $C_2^{12}.D_{12}$

Order 73728: $C_2^8.C_6^2.C_2^3$ x 2, $C_2^8.C_3:S_3.C_2^4$, $C_2^8.C_6^2.C_2^3$

Order 65536: $C_2^{10}.C_2^5.C_2$

Order 36864: $C_2^8.C_6^2.C_2^2$, $C_2^8.C_6^2.C_2^2$, $C_2^6.A_4^2:C_4$

Order 24576: $C_2^{12}.C_6$

Order 18432: $C_2^6.\PSOPlus(4,3)$ x 2, $C_2^8.C_6^2.C_2$

Order 12288: $C_2^{12}.C_3$

Order 9216: $C_2^{10}:C_3^2$, $C_2^5.\PSOPlus(4,3)$

Order 8192: $C_2^{13}$

Order 6912: $(C_2^3\times C_6^2):D_{12}$

Order 4608: $C_2^4:\PSOPlus(4,3)$ x 2, $C_2^9:C_3^2$

Order 4096: $C_2^{12}$

Order 2304: $C_2^4:A_4^2$

Order 2048: $C_2^{11}$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 32: $C_2^5$

Order 27: $C_3^3$

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1769472: $C_2\times C_2^{10}.C_6^2:D_{12}$ ($C_1$)

Order 884736: $C_2^{10}.C_6^2:D_{12}$ ($C_2$) x 4, $C_2^{12}.C_3^3.C_2^3$ ($C_2$), $C_2^{12}.C_6:S_3^2$ ($C_2$), $C_2\times C_2^{10}.C_6^2:C_{12}$ ($C_2$)

Order 442368: $C_2^{12}.C_3^2.D_6$ ($C_2^2$) x 2, $C_2^{10}.C_6^2:C_{12}$ ($C_2^2$) x 2, $C_2^{10}.A_4:S_3^2$ ($C_2^2$) x 2, $C_2^8.C_6^2.A_4.C_2^2$ ($C_2^2$)

Order 294912: $C_2^8.C_6^2.C_2^4.C_2$ ($S_3$)

Order 221184: $C_2^8.C_6^2.A_4.C_2$ ($C_2^3$), $C_2^8.C_6^2.A_4.C_2$ ($D_4$), $C_2^{12}.C_3^3.C_2$ ($D_4$)

Order 147456: $C_2^8.A_4^2:C_4$ ($D_6$) x 2, $C_2^8.C_6^2.C_2^4$ ($D_6$)

Order 110592: $C_2^{12}.C_3^3$ ($C_2\times D_4$)

Order 73728: $C_2^8.C_6^2.C_2^3$ ($D_{12}$), $C_2^8.C_6^2.C_2^3$ ($C_2\times D_6$), $C_2^8.C_3:S_3.C_2^4$ ($S_4$), $C_2^8.C_6^2.C_2^3$ ($D_{12}$)

Order 36864: $C_2^6.A_4^2:C_4$ ($C_2\times S_4$) x 2, $C_2^8.C_6^2.C_2^2$ ($C_2\times S_4$), $C_2^8.C_6^2.C_2^2$ ($C_2\times D_{12}$)

Order 24576: $C_2^{12}.C_6$ ($\SOPlus(4,2)$)

Order 18432: $C_2^8.C_6^2.C_2$ ($C_4:S_4$), $C_2^6.\PSOPlus(4,3)$ ($C_2^2\times S_4$), $C_2^6.\PSOPlus(4,3)$ ($C_4:S_4$)

Order 12288: $C_2^{12}.C_3$ ($S_3^2:C_2^2$)

Order 9216: $C_2^{10}:C_3^2$ ($C_2^3:D_{12}$), $C_2^5.\PSOPlus(4,3)$ ($C_2^3:S_4$)

Order 8192: $C_2^{13}$ ($C_3^2:D_{12}$)

Order 4608: $C_2^9:C_3^2$ ($C_2^4:D_{12}$), $C_2^4:\PSOPlus(4,3)$ ($C_2^4:D_{12}$), $C_2^4:\PSOPlus(4,3)$ ($C_2^4:S_4$)

Order 4096: $C_2^{12}$ ($C_2\times C_3^2:D_{12}$)

Order 2304: $C_2^4:A_4^2$ ($C_2^5:D_{12}$)

Order 2048: $C_2^{11}$ ($C_6^2:D_{12}$)

Order 1024: $C_2^{10}$ ($C_2\times C_6^2:D_{12}$)

Order 512: $C_2^9$ ($(C_2^2\times C_6^2):D_{12}$)

Order 256: $C_2^8$ ($(C_2^3\times C_6^2):D_{12}$)

Order 32: $C_2^5$ ($C_2^4:A_4^2.D_{12}$)

Order 16: $C_2^4$ ($C_2^9.C_3^2:D_{12}$)

Order 8: $C_2^3$ ($C_2^8.C_6^2:D_{12}$)

Order 4: $C_2^2$ ($C_2^9.C_6^2:D_{12}$)

Order 2: $C_2$ ($C_2^{10}.C_6^2:D_{12}$)

Order 1: $C_1$ ($C_2\times C_2^{10}.C_6^2:D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1769472: $C_2\times C_2^{10}.C_6^2:D_{12}$ ($C_1$)

Order 884736: $C_2^{12}.C_3^3.C_2^3$ ($C_2$), $C_2^{12}.C_6:S_3^2$ ($C_2$), $C_2\times C_2^{10}.C_6^2:C_{12}$ ($C_2$), $C_2^{10}.C_6^2:D_{12}$ ($C_2$)

Order 442368: $C_2^8.C_6^2.A_4.C_2^2$ ($C_2^2$), $C_2^{12}.C_3^2.D_6$ ($C_2^2$), $C_2^{10}.C_6^2:C_{12}$ ($C_2^2$), $C_2^{10}.A_4:S_3^2$ ($C_2^2$)

Order 294912: $C_2^8.C_6^2.C_2^4.C_2$ ($S_3$)

Order 221184: $C_2^8.C_6^2.A_4.C_2$ ($C_2^3$), $C_2^8.C_6^2.A_4.C_2$ ($D_4$), $C_2^{12}.C_3^3.C_2$ ($D_4$)

Order 147456: $C_2^8.C_6^2.C_2^4$ ($D_6$), $C_2^8.A_4^2:C_4$ ($D_6$)

Order 110592: $C_2^{12}.C_3^3$ ($C_2\times D_4$)

Order 73728: $C_2^8.C_6^2.C_2^3$ ($D_{12}$), $C_2^8.C_6^2.C_2^3$ ($C_2\times D_6$), $C_2^8.C_3:S_3.C_2^4$ ($S_4$), $C_2^8.C_6^2.C_2^3$ ($D_{12}$)

Order 36864: $C_2^8.C_6^2.C_2^2$ ($C_2\times S_4$), $C_2^8.C_6^2.C_2^2$ ($C_2\times D_{12}$), $C_2^6.A_4^2:C_4$ ($C_2\times S_4$)

Order 24576: $C_2^{12}.C_6$ ($\SOPlus(4,2)$)

Order 18432: $C_2^8.C_6^2.C_2$ ($C_4:S_4$), $C_2^6.\PSOPlus(4,3)$ ($C_2^2\times S_4$), $C_2^6.\PSOPlus(4,3)$ ($C_4:S_4$)

Order 12288: $C_2^{12}.C_3$ ($S_3^2:C_2^2$)

Order 9216: $C_2^{10}:C_3^2$ ($C_2^3:D_{12}$), $C_2^5.\PSOPlus(4,3)$ ($C_2^3:S_4$)

Order 8192: $C_2^{13}$ ($C_3^2:D_{12}$)

Order 4608: $C_2^9:C_3^2$ ($C_2^4:D_{12}$), $C_2^4:\PSOPlus(4,3)$ ($C_2^4:D_{12}$), $C_2^4:\PSOPlus(4,3)$ ($C_2^4:S_4$)

Order 4096: $C_2^{12}$ ($C_2\times C_3^2:D_{12}$)

Order 2304: $C_2^4:A_4^2$ ($C_2^5:D_{12}$)

Order 2048: $C_2^{11}$ ($C_6^2:D_{12}$)

Order 1024: $C_2^{10}$ ($C_2\times C_6^2:D_{12}$)

Order 512: $C_2^9$ ($(C_2^2\times C_6^2):D_{12}$)

Order 256: $C_2^8$ ($(C_2^3\times C_6^2):D_{12}$)

Order 32: $C_2^5$ ($C_2^4:A_4^2.D_{12}$)

Order 16: $C_2^4$ ($C_2^9.C_3^2:D_{12}$)

Order 8: $C_2^3$ ($C_2^8.C_6^2:D_{12}$)

Order 4: $C_2^2$ ($C_2^9.C_6^2:D_{12}$)

Order 2: $C_2$ ($C_2^{10}.C_6^2:D_{12}$)

Order 1: $C_1$ ($C_2\times C_2^{10}.C_6^2:D_{12}$)

Series

Derived series

$C_2\times C_2^{10}.C_6^2:D_{12}$

$\rhd$

$C_2^8.C_6^2.A_4.C_2$

$\rhd$

$C_2^8.C_6^2.C_2^2$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$C_2\times C_2^{10}.C_6^2:D_{12}$

$\rhd$

$C_2^{12}.C_3^3.C_2^3$

$\rhd$

$C_2^8.C_6^2.A_4.C_2^2$

$\rhd$

$C_2^8.C_6^2.A_4.C_2$

$\rhd$

$C_2^{12}.C_3^3$

$\rhd$

$C_2^8.C_6^2.C_2^2$

$\rhd$

$C_2^{10}:C_3^2$

$\rhd$

$C_2^4:A_4^2$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Lower central series

$C_2\times C_2^{10}.C_6^2:D_{12}$

$\rhd$

$C_2^8.C_6^2.A_4.C_2$

$\rhd$

$C_2^{12}.C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

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

Character theory

Complex character table

The $384 \times 384$ character table is not available for this group.

Rational character table

The $360 \times 360$ rational character table is not available for this group.