Abstract group 2654208.cn: $C_2^9.C_6:S_3^3:C_4$

• Label: 2654208.cn
• Order: \( 2^{15} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{18} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2^9.C_6:S_3^3:C_4$
Order:	\(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^8.C_3^4.C_2^6.C_2^4$, of order \(21233664\)\(\medspace = 2^{18} \cdot 3^{4} \)
Composition factors:	$C_2$ x 15, $C_3$ x 4
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	24
Elements	1	11967	6560	357696	204384	580608	829440	663552	2654208
Conjugacy classes	1	36	6	101	69	44	67	20	344
Divisions	1	36	6	77	69	24	49	10	272
Autjugacy classes	1	30	6	56	53	10	36	3	195

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:	${\langle a, b, c, d, e, f, g, h, i, j, k, l \mid c^{12}=d^{6}=e^{6}=f^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,12,6,7)(2,11,5,8)(3,9,4,10)(13,17,21,14,18,22)(19,23,20,24), (1,15,7,22,5,19,3,17,9,23,11,13,2,16,8,21,6,20,4,18,10,24,12,14) \!\cdots\! \rangle$
Transitive group:	24T22610	24T22611			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:	$(C_2^9.C_3:S_3^3)$ . $D_4$ (2)	$(C_2^9.C_6:S_3^3)$ . $C_4$	$C_2^9$ . $(C_6:S_3^3:C_4)$	$C_2^{10}$ . $(C_3:S_3^3:C_4)$	all 40

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

Homology

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

Subgroups

There are 54 normal subgroups (44 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $A_4^2\wr C_2.D_4.C_4$
Commutator:	$G' \simeq$ $C_2^9.C_3^2:S_3^2$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $A_4^2\wr C_2.D_4.C_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^{10}$	$G/\operatorname{Fit} \simeq$ $C_3:S_3^3:C_4$
Radical:	$R \simeq$ $C_2^9.C_6:S_3^3:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^9$	$G/\operatorname{soc} \simeq$ $C_6:S_3^3:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2^5.C_2^4$
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 18 (order at least 147456) are shown, as well as all normal subgroups of any index.

Order 2654208: $C_2^9.C_6:S_3^3:C_4$

Order 1327104: $C_2^9.C_3:S_3^3:C_4$ x 2, $C_2^9.C_3:S_3^3:C_4$ x 2, $C_2^8.C_3^4.C_2^4.C_2^2$, $C_2^2.(A_4^2\wr C_2.D_4)$, $C_2^{10}.C_3^4:(C_2\times D_4)$

Order 663552: $C_2^8.C_3^4.C_2.C_4^2$ x 3, $C_2.A_4^2\wr C_2.D_4$ x 3, $C_2.A_4^2\wr C_2.D_4$ x 2, $C_2.A_4^2\wr C_2.D_4$ x 2, $A_4^2.S_4\wr C_2.C_4$ x 2, $C_2^8.C_3^4.C_2^4.C_2$, $C_2^8.C_3^4.C_2^4.C_2$, $C_2^9.C_6:S_3^3$, $A_4^2.S_4^2:D_4$, $C_2^{10}.C_3^4:D_4$, $C_2^{10}.C_3^4:(C_2\times C_4)$, $C_2.A_4\wr C_4.C_2^2$

Order 331776: $C_2^8.C_3^4.C_2^3.C_2$ x 9, $C_2^9.C_3:S_3^3$ x 5, $C_2^8.C_3^4.C_2^3.C_2$ x 2, $A_4^2.S_4^2:C_2^2$ x 2, $C_2^9.C_3^4:D_4$ x 2, $C_2^9.C_3^4:D_4$ x 2, $C_2^9.C_3^4:D_4$ x 2, $C_2.A_4^2\wr C_2.C_4$ x 2, $C_2.A_4\wr C_4.C_2$ x 2, $C_2^8.C_3^4.C_2^3.C_2$, $C_2^6.C_3^3.A_4.C_2^3.C_2$, $C_2^8.C_3^4.C_2^4$, $C_2^9.C_3:S_3^3$, $C_2^9.C_3:S_3^3$, $C_2^9.C_3:S_3^3$, $C_2^9.C_3:S_3^3$, $C_2^2.A_4\wr C_4$, $C_2^{10}.C_3\wr C_2^2$, $A_4^2.C_2\wr S_3^2$, $C_2^{10}.C_3^4:C_4$, $C_2^8.C_3^4:C_4^2$, $C_2^8.C_3^4:\OD_{16}$

Order 221184: $C_2^6.C_3^3.C_2^4.C_2^3$, $C_2^8.C_3.D_6^2.C_2$, $C_2^4:A_4^2.(S_3\times \OD_{16})$, $C_2^{10}.S_3^3$

Order 165888: $C_2^8.C_3^4.C_2^2.C_2$ x 10, $C_2^8.C_3^4.C_2^3$ x 7, $C_2^9.C_3^2:S_3^2$ x 7, $C_2^6.C_3^3.A_4.C_2^2.C_2$ x 6, $C_2^9.C_3\wr C_2^2$ x 4, $A_4^2.S_4\wr C_2$ x 4, $C_2^8.C_3^4.C_2^2.C_2$ x 2, $C_2.A_4\wr C_4$ x 2, $A_4^2\wr C_2.C_4$ x 2, $C_2^9.C_3^4:C_4$ x 2, $C_2^8.C_3^4.C_2^3$, $A_4^2.A_4^2.C_2^3$, $C_2^6.C_3^4.C_2^4.C_2$, $C_2^9.C_3\wr C_2^2$, $C_2^9.C_3\wr C_2^2$, $A_4^2.A_4^2:D_4$, $A_4^2\wr C_2.C_2^2$, $C_2^{10}.C_3^2\wr C_2$

Order 147456: $A_4^2.C_2^3.C_2^5.C_2^2$, $C_2^4.S_4^2:\OD_{16}$, $C_2^5.S_4^2:C_2^3$

Order 82944: $C_2^8.C_3^3.D_6$ x 4, $C_2^8.C_3^3.D_6$ x 2, $C_2^8.C_3^4.C_2^2$

Order 41472: $C_2^8.C_3^4.C_2$ x 2, $C_2^8.C_3^4.C_2$

Order 32768: $C_2^6.C_2^5.C_2^4$

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

Order 10368: $C_6^2:S_3^2.D_4$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 81: $C_3^4$

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 147456) are shown, as well as all normal subgroups of any index.

Order 2654208: $C_2^9.C_6:S_3^3:C_4$

Order 1327104: $C_2^8.C_3^4.C_2^4.C_2^2$, $C_2^9.C_3:S_3^3:C_4$, $C_2^9.C_3:S_3^3:C_4$, $C_2^2.(A_4^2\wr C_2.D_4)$, $C_2^{10}.C_3^4:(C_2\times D_4)$

Order 663552: $C_2^8.C_3^4.C_2.C_4^2$ x 2, $C_2.A_4^2\wr C_2.D_4$ x 2, $C_2.A_4^2\wr C_2.D_4$ x 2, $C_2.A_4^2\wr C_2.D_4$ x 2, $A_4^2.S_4\wr C_2.C_4$ x 2, $C_2^8.C_3^4.C_2^4.C_2$, $C_2^8.C_3^4.C_2^4.C_2$, $C_2^9.C_6:S_3^3$, $A_4^2.S_4^2:D_4$, $C_2^{10}.C_3^4:D_4$, $C_2^{10}.C_3^4:(C_2\times C_4)$, $C_2.A_4\wr C_4.C_2^2$

Order 331776: $C_2^9.C_3:S_3^3$ x 5, $C_2^8.C_3^4.C_2^3.C_2$ x 4, $C_2^8.C_3^4.C_2^3.C_2$ x 2, $A_4^2.S_4^2:C_2^2$ x 2, $C_2^9.C_3^4:D_4$ x 2, $C_2^9.C_3^4:D_4$ x 2, $C_2^9.C_3^4:D_4$ x 2, $C_2.A_4^2\wr C_2.C_4$ x 2, $C_2^8.C_3^4.C_2^3.C_2$, $C_2^6.C_3^3.A_4.C_2^3.C_2$, $C_2^8.C_3^4.C_2^4$, $C_2^9.C_3:S_3^3$, $C_2^9.C_3:S_3^3$, $C_2^9.C_3:S_3^3$, $C_2^9.C_3:S_3^3$, $C_2^2.A_4\wr C_4$, $C_2^{10}.C_3\wr C_2^2$, $A_4^2.C_2\wr S_3^2$, $C_2^{10}.C_3^4:C_4$, $C_2^8.C_3^4:C_4^2$, $C_2^8.C_3^4:\OD_{16}$, $C_2.A_4\wr C_4.C_2$

Order 221184: $C_2^6.C_3^3.C_2^4.C_2^3$, $C_2^8.C_3.D_6^2.C_2$, $C_2^4:A_4^2.(S_3\times \OD_{16})$, $C_2^{10}.S_3^3$

Order 165888: $C_2^9.C_3^2:S_3^2$ x 7, $C_2^8.C_3^4.C_2^3$ x 6, $C_2^9.C_3\wr C_2^2$ x 4, $C_2^8.C_3^4.C_2^2.C_2$ x 3, $A_4^2.S_4\wr C_2$ x 3, $C_2^8.C_3^4.C_2^2.C_2$ x 2, $C_2^6.C_3^3.A_4.C_2^2.C_2$ x 2, $C_2^9.C_3^4:C_4$ x 2, $C_2^8.C_3^4.C_2^3$, $A_4^2.A_4^2.C_2^3$, $C_2^6.C_3^4.C_2^4.C_2$, $C_2.A_4\wr C_4$, $C_2^9.C_3\wr C_2^2$, $C_2^9.C_3\wr C_2^2$, $A_4^2\wr C_2.C_4$, $A_4^2.A_4^2:D_4$, $A_4^2\wr C_2.C_2^2$, $C_2^{10}.C_3^2\wr C_2$

Order 147456: $A_4^2.C_2^3.C_2^5.C_2^2$, $C_2^4.S_4^2:\OD_{16}$, $C_2^5.S_4^2:C_2^3$

Order 82944: $C_2^8.C_3^3.D_6$ x 3, $C_2^8.C_3^3.D_6$ x 2, $C_2^8.C_3^4.C_2^2$

Order 41472: $C_2^8.C_3^4.C_2$ x 2, $C_2^8.C_3^4.C_2$

Order 32768: $C_2^6.C_2^5.C_2^4$

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

Order 10368: $C_6^2:S_3^2.D_4$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 81: $C_3^4$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2654208: $C_2^9.C_6:S_3^3:C_4$ ($C_1$)

Order 1327104: $C_2^9.C_3:S_3^3:C_4$ ($C_2$) x 2, $C_2^9.C_3:S_3^3:C_4$ ($C_2$) x 2, $C_2^8.C_3^4.C_2^4.C_2^2$ ($C_2$), $C_2^2.(A_4^2\wr C_2.D_4)$ ($C_2$), $C_2^{10}.C_3^4:(C_2\times D_4)$ ($C_2$)

Order 663552: $C_2^8.C_3^4.C_2.C_4^2$ ($C_2^2$) x 2, $C_2.A_4^2\wr C_2.D_4$ ($C_2^2$) x 2, $C_2^8.C_3^4.C_2^4.C_2$ ($C_2^2$), $C_2^9.C_6:S_3^3$ ($C_4$), $C_2.A_4^2\wr C_2.D_4$ ($C_2^2$), $C_2.A_4^2\wr C_2.D_4$ ($C_4$), $C_2.A_4^2\wr C_2.D_4$ ($C_2^2$), $C_2.A_4^2\wr C_2.D_4$ ($C_4$), $C_2^{10}.C_3^4:(C_2\times C_4)$ ($C_4$)

Order 331776: $C_2^9.C_3:S_3^3$ ($D_4$) x 2, $C_2.A_4^2\wr C_2.C_4$ ($C_2\times C_4$) x 2, $C_2^8.C_3^4.C_2^3.C_2$ ($D_4$), $C_2^8.C_3^4.C_2^3.C_2$ ($C_2\times C_4$), $C_2^8.C_3^4.C_2^4$ ($D_4$), $C_2^9.C_3:S_3^3$ ($C_2\times C_4$), $C_2^9.C_3:S_3^3$ ($C_2\times C_4$), $C_2^9.C_3:S_3^3$ ($C_2^3$), $C_2^9.C_3:S_3^3$ ($C_2\times C_4$)

Order 165888: $C_2^9.C_3^2:S_3^2$ ($C_2^2:C_4$) x 2, $C_2^8.C_3^4.C_2^3$ ($C_2^2:C_4$), $C_2^8.C_3^4.C_2^3$ ($C_2^2:C_4$), $C_2^8.C_3^4.C_2^3$ ($C_2\times D_4$), $C_2^9.C_3^2:S_3^2$ ($C_2\times D_4$), $C_2^9.C_3^2:S_3^2$ ($C_2^2\times C_4$)

Order 82944: $C_2^8.C_3^3.D_6$ ($C_4\wr C_2$) x 2, $C_2^8.C_3^3.D_6$ ($Q_{16}:C_2$), $C_2^8.C_3^3.D_6$ ($D_8:C_2$), $C_2^8.C_3^3.D_6$ ($C_2^3:C_4$), $C_2^8.C_3^3.D_6$ ($C_2^3:C_4$), $C_2^8.C_3^4.C_2^2$ ($C_2^3:C_4$)

Order 41472: $C_2^8.C_3^4.C_2$ ($C_2^4:C_4$), $C_2^8.C_3^4.C_2$ ($C_2^3.D_4$), $C_2^8.C_3^4.C_2$ ($C_4^2:C_2^2$)

Order 20736: $C_2^8.C_3^4$ ($C_2^4.D_4$)

Order 1024: $C_2^{10}$ ($C_3:S_3^3:C_4$)

Order 512: $C_2^9$ ($C_6:S_3^3:C_4$)

Order 256: $C_2^8$ ($C_6^2:S_3^2.D_4$)

Order 4: $C_2^2$ ($A_4^2:\POPlus(4,3).D_4$)

Order 2: $C_2$ ($A_4^2\wr C_2.D_4.C_4$)

Order 1: $C_1$ ($C_2^9.C_6:S_3^3:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2654208: $C_2^9.C_6:S_3^3:C_4$ ($C_1$)

Order 1327104: $C_2^8.C_3^4.C_2^4.C_2^2$ ($C_2$), $C_2^9.C_3:S_3^3:C_4$ ($C_2$), $C_2^9.C_3:S_3^3:C_4$ ($C_2$), $C_2^2.(A_4^2\wr C_2.D_4)$ ($C_2$), $C_2^{10}.C_3^4:(C_2\times D_4)$ ($C_2$)

Order 663552: $C_2^8.C_3^4.C_2.C_4^2$ ($C_2^2$), $C_2^8.C_3^4.C_2^4.C_2$ ($C_2^2$), $C_2^9.C_6:S_3^3$ ($C_4$), $C_2.A_4^2\wr C_2.D_4$ ($C_2^2$), $C_2.A_4^2\wr C_2.D_4$ ($C_2^2$), $C_2.A_4^2\wr C_2.D_4$ ($C_4$), $C_2.A_4^2\wr C_2.D_4$ ($C_2^2$), $C_2.A_4^2\wr C_2.D_4$ ($C_4$), $C_2^{10}.C_3^4:(C_2\times C_4)$ ($C_4$)

Order 331776: $C_2^9.C_3:S_3^3$ ($D_4$) x 2, $C_2.A_4^2\wr C_2.C_4$ ($C_2\times C_4$) x 2, $C_2^8.C_3^4.C_2^3.C_2$ ($D_4$), $C_2^8.C_3^4.C_2^3.C_2$ ($C_2\times C_4$), $C_2^8.C_3^4.C_2^4$ ($D_4$), $C_2^9.C_3:S_3^3$ ($C_2\times C_4$), $C_2^9.C_3:S_3^3$ ($C_2\times C_4$), $C_2^9.C_3:S_3^3$ ($C_2^3$), $C_2^9.C_3:S_3^3$ ($C_2\times C_4$)

Order 165888: $C_2^9.C_3^2:S_3^2$ ($C_2^2:C_4$) x 2, $C_2^8.C_3^4.C_2^3$ ($C_2^2:C_4$), $C_2^8.C_3^4.C_2^3$ ($C_2^2:C_4$), $C_2^8.C_3^4.C_2^3$ ($C_2\times D_4$), $C_2^9.C_3^2:S_3^2$ ($C_2\times D_4$), $C_2^9.C_3^2:S_3^2$ ($C_2^2\times C_4$)

Order 82944: $C_2^8.C_3^3.D_6$ ($Q_{16}:C_2$), $C_2^8.C_3^3.D_6$ ($D_8:C_2$), $C_2^8.C_3^3.D_6$ ($C_4\wr C_2$), $C_2^8.C_3^3.D_6$ ($C_2^3:C_4$), $C_2^8.C_3^3.D_6$ ($C_2^3:C_4$), $C_2^8.C_3^4.C_2^2$ ($C_2^3:C_4$)

Order 41472: $C_2^8.C_3^4.C_2$ ($C_2^4:C_4$), $C_2^8.C_3^4.C_2$ ($C_2^3.D_4$), $C_2^8.C_3^4.C_2$ ($C_4^2:C_2^2$)

Order 20736: $C_2^8.C_3^4$ ($C_2^4.D_4$)

Order 1024: $C_2^{10}$ ($C_3:S_3^3:C_4$)

Order 512: $C_2^9$ ($C_6:S_3^3:C_4$)

Order 256: $C_2^8$ ($C_6^2:S_3^2.D_4$)

Order 4: $C_2^2$ ($A_4^2:\POPlus(4,3).D_4$)

Order 2: $C_2$ ($A_4^2\wr C_2.D_4.C_4$)

Order 1: $C_1$ ($C_2^9.C_6:S_3^3:C_4$)

Series

Derived series

$C_2^9.C_6:S_3^3:C_4$

$\rhd$

$C_2^9.C_3^2:S_3^2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$C_2^9.C_6:S_3^3:C_4$

$\rhd$

$C_2^8.C_3^4.C_2^4.C_2^2$

$\rhd$

$C_2^8.C_3^4.C_2^4.C_2$

$\rhd$

$C_2^9.C_3:S_3^3$

$\rhd$

$C_2^9.C_3^2:S_3^2$

$\rhd$

$C_2^8.C_3^3.D_6$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Lower central series

$C_2^9.C_6:S_3^3:C_4$

$\rhd$

$C_2^9.C_3^2:S_3^2$

$\rhd$

$C_2^8.C_3^3.D_6$

$\rhd$

$C_2^8.C_3^4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

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 $344 \times 344$ character table is not available for this group.

Rational character table

The $272 \times 272$ rational character table is not available for this group.