Abstract group 393216.cq: $C_2^8.C_2\wr S_4$

• Label: 393216.cq
• Order: \( 2^{17} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{23} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2^8.C_2\wr S_4$
Order:	\(393216\)\(\medspace = 2^{17} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^8.C_2^6.C_6.C_2^4.C_2^4$, of order \(25165824\)\(\medspace = 2^{23} \cdot 3 \)
Composition factors:	$C_2$ x 17, $C_3$
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	7871	8192	143680	90112	110592	32768	393216
Conjugacy classes	1	100	1	223	7	40	2	374
Divisions	1	100	1	219	6	30	1	358
Autjugacy classes	1	56	1	91	4	10	1	164

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	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:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m \mid c^{2}=d^{4}=e^{4}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,10,5,13,22,18,2,9,6,14,21,17)(3,23,8,15,11,20,4,24,7,16,12,19), (1,19,2,20) \!\cdots\! \rangle$
Transitive group:	24T19702	24T19941	24T20011		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:	$C_2^8$ . $(C_2\wr S_4)$	$(C_2^9.C_2^5)$ . $S_4$ (5)	$C_2^9$ . $(C_2^5:S_4)$	$C_2^9$ . $(C_2^5:S_4)$	all 46

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$2$

Subgroups

There are 94 normal subgroups (50 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^8.C_2^6.D_6$
Commutator:	$G' \simeq$ $C_2^9.C_2^6.C_3$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2^9$	$G/\Phi \simeq$ $C_2^5:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^{11}.C_2^5$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^8.C_2\wr S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^5.C_2\wr S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^5.C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 8 (order at least 49152) are shown, as well as all normal subgroups of any index.

Order 393216: $C_2^8.C_2\wr S_4$

Order 196608: $C_2^9.C_2^6.C_6$, $C_2^6.C_2^7:S_4$, $C_2^6.C_2^7:S_4$

Order 131072: $C_2^9.C_2^5.C_2^3$

Order 98304: $C_2^6.C_2\wr S_4$ x 2, $C_2^{10}.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^7.C_2^4.C_6.C_2^3$, $C_2^9.C_2^6.C_3$, $C_2^6.C_2\wr S_4$, $C_2^6.C_2\wr S_4$

Order 65536: $C_2^9.C_2^4.C_2^3$ x 2, $C_2^9.C_2^6.C_2$ x 2, $C_2^9.C_2^4.C_2^3$ x 2, $C_2^8.C_2^6.C_2^2$ x 2, $C_2^9.C_2^4.C_2^3$ x 2, $C_2^9.C_2^4.C_2^3$ x 2, $C_2^9.C_2^4.C_2^3$ x 2, $C_2^{10}.C_2^5.C_2$ x 2, $C_2^9.C_2^4.C_2^3$ x 2, $C_2^9.C_2^4.C_2^3$ x 2, $C_2^7.C_2^6.C_2^3$ x 2, $C_2^9.C_2^5.C_2^2$ x 2, $C_2^8.C_2^5.C_2^3$, $C_2^6.C_2^6.C_2^4$, $C_2^9.C_2^5.C_2^2$, $C_2^8.C_2^5.C_2^3$, $C_2^8.C_2^5.C_2^3$, $C_2^{11}.C_2^5$, $C_2^8.C_2^5.C_2^3$

Order 49152: $C_2^7.C_4^2:S_4$ x 3, $C_2^5.C_2\wr S_4$ x 3, $C_2^5.C_2\wr S_4$ x 2, $C_2^5.C_2\wr S_4$ x 2, $C_2^5.C_2\wr S_4$ x 2, $C_2^7.C_2\wr S_3$ x 2, $C_2^7.C_4^2:S_4$ x 2, $C_2^7.C_2\wr S_3$ x 2, $C_2^5.C_2\wr S_4$ x 2, $C_2^6.C_2\wr A_4$ x 2, $C_2^6.C_2\wr A_4$ x 2, $C_2^6.C_2\wr A_4$ x 2, $C_2^7.C_4^2:S_4$ x 2, $C_2^6.C_2\wr A_4$ x 2, $C_2^{11}.A_4.C_2$, $C_2^9.C_2^4.C_6$, $C_2^{10}.A_4.C_2^2$, $C_2^8.C_2^4.C_6.C_2$, $C_2^7.C_2^4.C_6.C_2^2$, $C_2^5.C_2\wr S_4$, $C_2^7.C_4^2:S_4$, $C_2^5.C_2^6:S_4$, $C_2^7.C_2^2\wr S_3$, $C_2^6.C_2\wr A_4$

Order 32768: $C_2^9.C_2^6$

Order 16384: $C_2^9.C_2^5$ x 2, $C_2^9.C_2^5$ x 2, $C_2^8.C_2^6$, $C_2^9.C_2^5$, $C_2^9.C_2^5$

Order 8192: $C_2^9.C_2^4$ x 3, $C_2^9.C_2^4$ x 3, $C_2^9.C_2^4$

Order 4096: $C_2^9.C_2^3$ x 2, $C_2^9.C_2^3$ x 2, $C_2^9.C_2^3$, $C_2^8.C_2^4$, $C_2^{11}.C_2$

Order 2048: $C_2^9.C_2^2$ x 3, $C_2^9.C_2^2$ x 3, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^{11}$

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

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

Order 256: $C_2^8$ x 6, $C_2^4\times C_4^2$ x 3, $D_4\times C_2^5$

Order 128: $C_2^7$ x 5, $C_2^5\times C_4$

Order 64: $C_2^6$ x 9

Order 32: $C_2^5$ x 4

Order 16: $C_2^4$ x 3

Order 8: $C_2^3$ x 3

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 8 (order at least 49152) are shown, as well as all normal subgroups of any index.

Order 393216: $C_2^8.C_2\wr S_4$

Order 196608: $C_2^9.C_2^6.C_6$, $C_2^6.C_2^7:S_4$, $C_2^6.C_2^7:S_4$

Order 131072: $C_2^9.C_2^5.C_2^3$

Order 98304: $C_2^7.C_2^4.C_6.C_2^3$, $C_2^9.C_2^6.C_3$, $C_2^6.C_2\wr S_4$, $C_2^6.C_2\wr S_4$, $C_2^6.C_2\wr S_4$, $C_2^{10}.\GL(2,\mathbb{Z}/4)$

Order 65536: $C_2^9.C_2^4.C_2^3$ x 2, $C_2^9.C_2^4.C_2^3$ x 2, $C_2^9.C_2^4.C_2^3$ x 2, $C_2^9.C_2^5.C_2^2$ x 2, $C_2^9.C_2^4.C_2^3$, $C_2^8.C_2^5.C_2^3$, $C_2^9.C_2^6.C_2$, $C_2^9.C_2^4.C_2^3$, $C_2^6.C_2^6.C_2^4$, $C_2^8.C_2^6.C_2^2$, $C_2^9.C_2^5.C_2^2$, $C_2^9.C_2^4.C_2^3$, $C_2^{10}.C_2^5.C_2$, $C_2^9.C_2^4.C_2^3$, $C_2^8.C_2^5.C_2^3$, $C_2^8.C_2^5.C_2^3$, $C_2^{11}.C_2^5$, $C_2^7.C_2^6.C_2^3$, $C_2^8.C_2^5.C_2^3$

Order 49152: $C_2^7.C_4^2:S_4$ x 2, $C_2^5.C_2\wr S_4$ x 2, $C_2^{11}.A_4.C_2$, $C_2^9.C_2^4.C_6$, $C_2^{10}.A_4.C_2^2$, $C_2^8.C_2^4.C_6.C_2$, $C_2^7.C_2^4.C_6.C_2^2$, $C_2^5.C_2\wr S_4$, $C_2^5.C_2\wr S_4$, $C_2^5.C_2\wr S_4$, $C_2^5.C_2\wr S_4$, $C_2^7.C_2\wr S_3$, $C_2^7.C_4^2:S_4$, $C_2^7.C_4^2:S_4$, $C_2^5.C_2^6:S_4$, $C_2^7.C_2^2\wr S_3$, $C_2^7.C_2\wr S_3$, $C_2^5.C_2\wr S_4$, $C_2^6.C_2\wr A_4$, $C_2^6.C_2\wr A_4$, $C_2^6.C_2\wr A_4$, $C_2^6.C_2\wr A_4$, $C_2^7.C_4^2:S_4$, $C_2^6.C_2\wr A_4$

Order 32768: $C_2^9.C_2^6$

Order 16384: $C_2^9.C_2^5$, $C_2^8.C_2^6$, $C_2^9.C_2^5$, $C_2^9.C_2^5$, $C_2^9.C_2^5$

Order 8192: $C_2^9.C_2^4$ x 2, $C_2^9.C_2^4$ x 2, $C_2^9.C_2^4$

Order 4096: $C_2^9.C_2^3$, $C_2^9.C_2^3$, $C_2^9.C_2^3$, $C_2^8.C_2^4$, $C_2^{11}.C_2$

Order 2048: $C_2^9.C_2^2$ x 2, $C_2^9.C_2^2$ x 2, $C_2^7.C_2^4$, $C_2^7.C_2^4$, $C_2^{11}$

Order 1024: $C_2^8.C_2^2$ x 2, $C_2^6.C_2^4$, $C_2^9.C_2$, $C_2^6.C_2^4$

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

Order 256: $C_2^8$ x 4, $C_2^4\times C_4^2$ x 2, $D_4\times C_2^5$

Order 128: $C_2^7$ x 4, $C_2^5\times C_4$

Order 64: $C_2^6$ x 5

Order 32: $C_2^5$ x 3

Order 16: $C_2^4$ x 2

Order 8: $C_2^3$ x 2

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 393216: $C_2^8.C_2\wr S_4$ ($C_1$)

Order 196608: $C_2^9.C_2^6.C_6$ ($C_2$), $C_2^6.C_2^7:S_4$ ($C_2$), $C_2^6.C_2^7:S_4$ ($C_2$)

Order 98304: $C_2^9.C_2^6.C_3$ ($C_2^2$)

Order 65536: $C_2^{11}.C_2^5$ ($S_3$)

Order 32768: $C_2^9.C_2^6$ ($D_6$)

Order 16384: $C_2^9.C_2^5$ ($S_4$) x 2, $C_2^9.C_2^5$ ($S_4$) x 2, $C_2^8.C_2^6$ ($S_4$), $C_2^9.C_2^5$ ($S_4$), $C_2^9.C_2^5$ ($S_4$)

Order 8192: $C_2^9.C_2^4$ ($C_2\times S_4$) x 3, $C_2^9.C_2^4$ ($C_2\times S_4$) x 3, $C_2^9.C_2^4$ ($C_2\times S_4$)

Order 4096: $C_2^9.C_2^3$ ($C_2^2:S_4$) x 2, $C_2^9.C_2^3$ ($C_2^2:S_4$) x 2, $C_2^9.C_2^3$ ($C_2^2:S_4$), $C_2^8.C_2^4$ ($C_2^2:S_4$), $C_2^{11}.C_2$ ($C_2^2:S_4$)

Order 2048: $C_2^9.C_2^2$ ($C_2^3:S_4$) x 3, $C_2^9.C_2^2$ ($C_2^3:S_4$) x 3, $C_2^7.C_2^4$ ($C_2^3:S_4$), $C_2^7.C_2^4$ ($C_2^3:S_4$), $C_2^{11}$ ($C_2^3:S_4$)

Order 1024: $C_2^8.C_2^2$ ($C_2^4:S_4$) x 2, $C_2^6.C_2^4$ ($C_4^2:S_4$), $C_2^8.C_2^2$ ($C_2^4:S_4$), $C_2^9.C_2$ ($C_2^4:S_4$), $C_2^6.C_2^4$ ($C_4^2:S_4$)

Order 512: $C_4^2.C_2^5$ ($C_2^5:S_4$) x 2, $C_2^6:D_4$ ($C_2^5:S_4$) x 2, $C_2^8.C_2$ ($C_2^5:S_4$), $C_2^8.C_2$ ($C_2^5:S_4$), $C_2^3\times C_4^3$ ($C_2^5:S_4$), $C_2^9$ ($C_2^5:S_4$), $C_2^9$ ($C_2^5:S_4$), $C_4^2.C_2^5$ ($C_2^5:S_4$), $C_4^2.C_2^5$ ($C_2^5:S_4$), $C_2^6:D_4$ ($C_2^5:S_4$), $C_2^6:D_4$ ($C_2^5:S_4$)

Order 256: $C_2^8$ ($C_2.C_2^6.D_6$) x 2, $C_2^8$ ($C_2\wr S_4$) x 2, $C_2^4\times C_4^2$ ($C_2\wr S_4$) x 2, $C_2^8$ ($C_2\times C_2^5.S_4$), $C_2^8$ ($C_2^6:S_4$), $D_4\times C_2^5$ ($C_2^6.S_4$), $C_2^4\times C_4^2$ ($C_2^6:S_4$)

Order 128: $C_2^7$ ($C_2^6.A_4.C_2^2$) x 2, $C_2^7$ ($C_2^6.S_4.C_2$), $C_2^7$ ($C_2^4.C_2^4.D_6$), $C_2^7$ ($C_2^7:S_4$), $C_2^5\times C_4$ ($C_2^7:S_4$)

Order 64: $C_2^6$ ($C_2^7.A_4.C_2^2$) x 2, $C_2^6$ ($C_2^7.A_4.C_2^2$) x 2, $C_2^6$ ($C_2^8.S_4$) x 2, $C_2^6$ ($C_2^8.S_4$) x 2, $C_2^6$ ($C_2^8.S_4$)

Order 32: $C_2^5$ ($C_2^6.C_2^4.D_6$) x 2, $C_2^5$ ($C_2^6.C_2^4.D_6$), $C_2^5$ ($C_2^8.A_4.C_2^2$)

Order 16: $C_2^4$ ($C_2^9.A_4.C_2^2$) x 2, $C_2^4$ ($C_2^9.A_4.C_2^2$)

Order 8: $C_2^3$ ($C_2^6.C_2^6.D_6$) x 2, $C_2^3$ ($C_2^5.C_2\wr S_4$)

Order 4: $C_2^2$ ($C_2^7.C_2^6.D_6$)

Order 2: $C_2$ ($C_2^8.C_2^6.D_6$)

Order 1: $C_1$ ($C_2^8.C_2\wr S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 393216: $C_2^8.C_2\wr S_4$ ($C_1$)

Order 196608: $C_2^9.C_2^6.C_6$ ($C_2$), $C_2^6.C_2^7:S_4$ ($C_2$), $C_2^6.C_2^7:S_4$ ($C_2$)

Order 98304: $C_2^9.C_2^6.C_3$ ($C_2^2$)

Order 65536: $C_2^{11}.C_2^5$ ($S_3$)

Order 32768: $C_2^9.C_2^6$ ($D_6$)

Order 16384: $C_2^9.C_2^5$ ($S_4$), $C_2^8.C_2^6$ ($S_4$), $C_2^9.C_2^5$ ($S_4$), $C_2^9.C_2^5$ ($S_4$), $C_2^9.C_2^5$ ($S_4$)

Order 8192: $C_2^9.C_2^4$ ($C_2\times S_4$) x 2, $C_2^9.C_2^4$ ($C_2\times S_4$) x 2, $C_2^9.C_2^4$ ($C_2\times S_4$)

Order 4096: $C_2^9.C_2^3$ ($C_2^2:S_4$), $C_2^9.C_2^3$ ($C_2^2:S_4$), $C_2^9.C_2^3$ ($C_2^2:S_4$), $C_2^8.C_2^4$ ($C_2^2:S_4$), $C_2^{11}.C_2$ ($C_2^2:S_4$)

Order 2048: $C_2^9.C_2^2$ ($C_2^3:S_4$) x 2, $C_2^9.C_2^2$ ($C_2^3:S_4$) x 2, $C_2^7.C_2^4$ ($C_2^3:S_4$), $C_2^7.C_2^4$ ($C_2^3:S_4$), $C_2^{11}$ ($C_2^3:S_4$)

Order 1024: $C_2^6.C_2^4$ ($C_4^2:S_4$), $C_2^8.C_2^2$ ($C_2^4:S_4$), $C_2^8.C_2^2$ ($C_2^4:S_4$), $C_2^9.C_2$ ($C_2^4:S_4$), $C_2^6.C_2^4$ ($C_4^2:S_4$)

Order 512: $C_2^8.C_2$ ($C_2^5:S_4$), $C_2^8.C_2$ ($C_2^5:S_4$), $C_2^3\times C_4^3$ ($C_2^5:S_4$), $C_2^9$ ($C_2^5:S_4$), $C_2^9$ ($C_2^5:S_4$), $C_4^2.C_2^5$ ($C_2^5:S_4$), $C_4^2.C_2^5$ ($C_2^5:S_4$), $C_4^2.C_2^5$ ($C_2^5:S_4$), $C_2^6:D_4$ ($C_2^5:S_4$), $C_2^6:D_4$ ($C_2^5:S_4$), $C_2^6:D_4$ ($C_2^5:S_4$)

Order 256: $C_2^8$ ($C_2.C_2^6.D_6$), $C_2^8$ ($C_2\times C_2^5.S_4$), $C_2^8$ ($C_2\wr S_4$), $C_2^8$ ($C_2^6:S_4$), $D_4\times C_2^5$ ($C_2^6.S_4$), $C_2^4\times C_4^2$ ($C_2\wr S_4$), $C_2^4\times C_4^2$ ($C_2^6:S_4$)

Order 128: $C_2^7$ ($C_2^6.S_4.C_2$), $C_2^7$ ($C_2^4.C_2^4.D_6$), $C_2^7$ ($C_2^6.A_4.C_2^2$), $C_2^7$ ($C_2^7:S_4$), $C_2^5\times C_4$ ($C_2^7:S_4$)

Order 64: $C_2^6$ ($C_2^7.A_4.C_2^2$), $C_2^6$ ($C_2^7.A_4.C_2^2$), $C_2^6$ ($C_2^8.S_4$), $C_2^6$ ($C_2^8.S_4$), $C_2^6$ ($C_2^8.S_4$)

Order 32: $C_2^5$ ($C_2^6.C_2^4.D_6$), $C_2^5$ ($C_2^6.C_2^4.D_6$), $C_2^5$ ($C_2^8.A_4.C_2^2$)

Order 16: $C_2^4$ ($C_2^9.A_4.C_2^2$), $C_2^4$ ($C_2^9.A_4.C_2^2$)

Order 8: $C_2^3$ ($C_2^6.C_2^6.D_6$), $C_2^3$ ($C_2^5.C_2\wr S_4$)

Order 4: $C_2^2$ ($C_2^7.C_2^6.D_6$)

Order 2: $C_2$ ($C_2^8.C_2^6.D_6$)

Order 1: $C_1$ ($C_2^8.C_2\wr S_4$)

Series

Derived series

$C_2^8.C_2\wr S_4$

$\rhd$

$C_2^9.C_2^6.C_3$

$\rhd$

$C_2^9.C_2^6$

$\rhd$

$C_2^9$

$\rhd$

$C_1$

Chief series

$C_2^8.C_2\wr S_4$

$\rhd$

$C_2^9.C_2^6.C_6$

$\rhd$

$C_2^9.C_2^6.C_3$

$\rhd$

$C_2^9.C_2^6$

$\rhd$

$C_2^9.C_2^4$

$\rhd$

$C_2^{11}$

$\rhd$

$C_2^9$

$\rhd$

$C_2^8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^8.C_2\wr S_4$

$\rhd$

$C_2^9.C_2^6.C_3$

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 1 larger groups in the database.

Character theory

Complex character table

The $374 \times 374$ character table is not available for this group.

Rational character table

The $358 \times 358$ rational character table is not available for this group.