Abstract group 663552.cu: $C_2^9.C_3^4:\OD_{16}$

• Label: 663552.cu
• Order: \( 2^{13} \cdot 3^{4} \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $20$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$C_2^9.C_3^4:\OD_{16}$
Order:	\(663552\)\(\medspace = 2^{13} \cdot 3^{4} \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	Group of order \(5308416\)\(\medspace = 2^{16} \cdot 3^{4} \)
Composition factors:	$C_2$ x 13, $C_3$ 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	16
Elements	1	5407	6560	87264	52832	290304	55296	165888	663552
Conjugacy classes	1	19	6	30	42	22	18	8	146
Divisions	1	19	6	27	42	9	18	2	124
Autjugacy classes	1	16	5	18	28	5	6	1	80

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:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid a^{8}=d^{6}=e^{6}=f^{6}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,2,4,8)(3,6,11,13)(5,9)(7,10)(12,15)(14,16)(18,19), (1,3,7,12)(2,5,10,11,14,6,4,9)(8,13,16,15)(17,18)(19,20)\rangle$
Transitive group:	24T20616	24T20619	36T33409		more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^9$ . $(C_3^4:\OD_{16})$	$(C_2^8.C_3^3.D_6)$ . $C_8$ (2)	$(C_2^9.C_3^4:C_4)$ . $C_4$	$(C_2^9.C_3^2:F_9)$ . $C_2$ (2)	all 15

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}^{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 20 normal subgroups (14 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2.C_4.C_2^3$

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 162 (order at least 4096) are shown, as well as all normal subgroups of any index.

Order 663552: $C_2^9.C_3^4:\OD_{16}$

Order 331776: $C_2^9.C_3^2:F_9$ x 2, $A_4^2.(A_4^2:C_4\times C_2^2)$

Order 165888: $C_2^8.C_3^2:F_9$ x 2, $C_2^9.C_3^4:C_4$ x 2, $A_4^2.(C_2\times A_4^2:C_4)$ x 2, $C_2^8.C_3^4.C_2^3$

Order 82944: $C_2^8.C_3^3.D_6$ x 4, $A_4^2:(A_4^2:C_4)$ x 3, $A_4^2.A_4^2.C_2^2$

Order 55296: $C_2^8.C_3^3.C_2^3$, $C_2^6.C_3^3.C_2.C_2^4$

Order 41472: $C_2^8.C_3^4.C_2$ x 3, $A_4^2.A_4^2.C_2$ x 2, $C_2^6.C_3^4.C_2^3$

Order 36864: $C_2^9.F_9$ x 2, $A_4^2.C_2^3.C_2.C_2^4$

Order 27648: $C_2^8.C_3^2.D_6$ x 9, $C_2^6.C_3^3.D_4.C_2$ x 6, $C_2^6.C_3^3.C_2^3.C_2$ x 4, $C_2^6.C_3^2.D_6.C_2^2$ x 2, $C_2^6.C_3^3.C_2^4$, $C_2^4:C_3.A_4^2.C_2^2$, $A_4^2.C_2^4.C_6.C_2$, $A_4^2.A_4.C_2^4$, $A_4.C_2^5.C_6^2.C_2$

Order 20736: $A_4^3.D_6$ x 4, $C_2^8.C_3^4$, $C_2^6.C_3^3.D_6$, $A_4^2.C_2^2.C_6^2$, $(C_3\times A_4).A_4^2.C_2^2$, $A_4^2:(C_6^2:C_4)$

Order 18432: $A_4^2.C_2^5:C_4$ x 8, $A_4^2.C_2\wr C_4.C_2$ x 4, $C_2^8.F_9$ x 4, $C_2^5.A_4^2:C_4$ x 3, $C_2^6.(C_6\times S_3).C_2^3$ x 2, $A_4^2.C_2^4.C_2^3$ x 2, $C_2^5.A_4^2:C_4$ x 2, $A_4^2.D_4^2.C_2$, $A_4^2.C_2^2.C_2.C_2^4$, $C_2^5.S_4^2$

Order 13824: $C_2^6.C_3^3.D_4$ x 12, $A_4^3.C_2^3$ x 11, $C_2^8.C_3^3.C_2$ x 9, $A_4^2.A_4.C_2^3$ x 5, $A_4.C_2^5.C_6^2$ x 4, $C_2^2:A_4^2.S_4$ x 4, $C_2^6.C_3^3.C_2^3$ x 3, $C_2^6.C_3^3.C_4.C_2$ x 2, $C_2^6.C_3^2.D_6.C_2$ x 2, $C_2^4:C_3.A_4^2.C_2$ x 2, $A_4^2.C_2^4.C_6$ x 2, $(C_3\times A_4^2).C_2.C_2^4$ x 2, $C_2^2.A_4^2:S_4$ x 2, $C_2^6.C_6^3$, $C_2^6.C_3^3.C_2^2.C_2$, $C_2^2\times A_4^2.A_4.C_2$, $C_2\times A_4.A_4^2.C_2^2$, $A_4^2.D_6.C_2^3$, $A_4.A_4^2.C_4.C_2$

Order 10368: $(C_3^2\times A_4^2):(C_2\times C_4)$ x 4, $C_2^4.C_3^4.C_2^3$ x 2, $A_4^3:S_3$ x 2, $A_4^3:S_3$ x 2, $A_4^3:C_6$ x 2, $C_2^5:(C_3^4:C_4)$ x 2, $C_2^6.C_3^4.C_2$

Order 9216: $A_4^2.C_2^3.C_2^3$ x 26, $A_4^2.D_4^2$ x 16, $C_2^6.C_3^2.D_4.C_2$ x 10, $C_2^6.(S_3\times C_3:D_4)$ x 10, $A_4^2.C_2^4.C_2^2$ x 9, $A_4^2:C_2\wr C_4$ x 8, $A_4^2:C_2\wr C_4$ x 8, $C_2^4.A_4^2:C_4$ x 6, $C_2^5.\PSOPlus(4,3)$ x 6, $A_4^2:\OD_{16}:C_2^2$ x 6, $A_4^2:C_2^4:C_4$ x 6, $C_2^6.D_6:S_3.C_2$ x 4, $C_2^4.A_4^2:C_4$ x 4, $(C_2^2:S_4)^2$ x 4, $A_4^2.(C_2^3.C_4).C_2$ x 3, $C_2^5.\PSOPlus(4,3)$ x 3, $C_2^5.\PSOPlus(4,3)$ x 3, $C_2^8.C_3.D_6$ x 2, $C_2^4.D_6:S_4.C_2$ x 2, $C_2^2.A_4^2.C_2^4$ x 2, $A_4.C_2^5.C_6.C_2^2$ x 2, $(C_2\times A_4^2).C_2^5$ x 2, $C_2^5.(A_4\times S_4)$ x 2, $C_2^4.S_4^2$ x 2, $C_2^6.C_6^2.C_2^2$, $C_2^6.(C_6\times S_3).C_2^2$, $C_2\times A_4^2.C_2^3.C_2^2$, $A_4^2.C_4:D_4.C_2$, $A_4^2.C_4.C_2^4$, $A_4^2.C_2^6$, $A_4^2.C_2^5.C_2$, $(C_2^4\times A_4).D_6.C_2^2$

Order 8192: $C_2^7.C_2.C_4.C_2^3$

Order 6912: $A_4^3:C_2^2$ x 20, $A_4:S_4^2$ x 20, $C_3:S_4\times \GL(2,\mathbb{Z}/4)$ x 16, $A_4^3:C_2^2$ x 12, $(C_2\times C_6):S_4^2$ x 8, $(C_2^2\times A_4^2):D_6$ x 8, $A_4.A_4^2.C_2^2$ x 7, $C_2^6.C_3^2.D_6$ x 6, $A_4^2.D_6.C_2^2$ x 4, $C_2^6.C_3^3.C_2^2$ x 3, $(C_3\times A_4^2).D_4.C_2$ x 3, $C_2^8.C_3^3$ x 2, $C_2^6.C_3^3.C_4$ x 2, $A_4^2.C_6.C_2^3$ x 2, $A_4.A_4^2.C_4$ x 2, $A_4.(C_6\times A_4).C_2^3$ x 2, $(C_3\times A_4^2).C_2^4$ x 2, $(C_3\times A_4^2).C_2.C_2^3$ x 2, $(C_3\times A_4).C_2^4.C_6.C_2$ x 2, $C_2^2\times C_3:S_4^2$ x 2, $C_2^2.A_4^3$ x 2, $C_2^4.C_6^2.C_6.C_2$, $C_2^2\times A_4^2.C_6.C_2$, $C_2^2.A_4^2.C_6.C_2$, $A_4^2.D_{12}.C_2$, $(C_2^4\times C_6).C_6^2.C_2$, $C_2^2\times A_4^2.D_6$, $C_2^2.A_4^3$

Order 6144: $A_4.C_2^5.C_2^4$, $C_2^9.D_6$

Order 5184: $C_3^2:S_4^2$ x 4, $(A_4\times C_6^2):D_6$ x 4, $(C_3^2\times A_4^2):C_4$ x 4, $(C_6^2\times A_4).D_6$ x 2, $C_3^2.A_4^2.C_2^2$, $A_4^2.C_6^2$, $A_4.(C_6\times S_3\times A_4)$, $(C_6^2\times A_4).C_6.C_2$, $C_3\times A_4^3$

Order 4608: $A_4^2.C_2^2\wr C_2$ x 48, $C_2^3:S_4^2$ x 38, $A_4^2.C_2^2:D_4$ x 32, $A_4^2:C_2^2\times D_4$ x 28, $A_4^2.C_2^3.C_2^2$ x 24, $(C_2\times A_4^2).C_2^4$ x 23, $C_2^3:S_4^2$ x 22, $A_4^2.C_2^4.C_2$ x 16, $(C_2^2\times A_4):\GL(2,\mathbb{Z}/4)$ x 16, $A_4^2.(C_4\times D_4)$ x 14, $C_2^6.C_6^2.C_2$ x 12, $C_2^4.(A_4\times S_4)$ x 12, $C_2^4.\PSOPlus(4,3)$ x 12, $A_4^2.C_2^2.C_2^3$ x 11, $(C_2^3\times A_4^2).C_4$ x 10, $C_2^3:(A_4^2:C_4)$ x 9, $C_2^6.C_3^2.C_4.C_2$ x 8, $A_4^2.C_4:D_4$ x 8, $A_4.C_2^5.C_6.C_2$ x 8, $C_2^3.S_4^2$ x 8, $C_2^4:\PSOPlus(4,3)$ x 8, $C_2^2\times A_4^2.C_2^3$ x 6, $C_2^9:C_3^2$ x 6, $C_2^4:\PSOPlus(4,3)$ x 6, $A_4^2:(C_2\times \OD_{16})$ x 6, $(C_2^3\times A_4^2):C_4$ x 6, $C_2^4.\PSOPlus(4,3)$ x 6, $C_2^6.C_3^2.D_4$ x 5, $C_2^4.D_6^2:C_2$ x 5, $C_2^8.C_3.S_3$ x 4, $C_2^6.(C_6\times S_3).C_2$ x 4, $C_2^3\times A_4^2.C_2^2$ x 4, $C_2^4:(A_4\times S_4)$ x 4, $(A_4\times C_2^4).S_4$ x 4, $A_4^2:C_2^3:C_4$ x 4, $C_2^5.C_6^2.C_2^2$ x 3, $C_2^4.D_6:S_4$ x 3, $C_2\times A_4^2.D_4.C_2$ x 3, $A_4^2.\OD_{16}.C_2$ x 3, $A_4^2.D_4.C_2^2$ x 3, $A_4^2.C_4^2.C_2$ x 3, $A_4^2.C_4.C_2^3$ x 3, $A_4^2.C_2^5$ x 3, $C_2^9:C_3^2$ x 3, $C_2^9:C_3^2$ x 3, $C_2^8.C_3.C_6$ x 2, $C_2^2.D_6^2.C_2^3$ x 2, $C_2\times A_4^2.C_2^4$ x 2, $A_4^2.C_4:C_4.C_2$ x 2, $A_4.C_2^4:C_3.C_4.C_2$ x 2, $(C_2^4\times A_4).D_6.C_2$ x 2, $C_2^4.(A_4\times S_4)$ x 2, $C_2^4.\PSOPlus(4,3)$ x 2, $C_2^5.(C_6\times C_{12}).C_2$, $C_2^4.(C_6\times S_3).C_2^3$, $C_2^2\times A_4.(D_4\times A_4)$, $C_2\times A_4^2.C_2^2.C_2^2$, $(C_2^3\times C_6).D_6.C_2^3$, $(C_2\times C_6\times A_4).C_2^5$, $C_2^3\times S_4^2$, $A_4^2:C_4\times C_2^3$

Order 4096: $C_2^6.C_2.C_4.C_2^3$ x 4, $C_2^7.C_4.C_2^3$ x 2, $C_2^6.C_2.C_2^5$

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 663552: $C_2^9.C_3^4:\OD_{16}$ ($C_1$)

Order 331776: $C_2^9.C_3^2:F_9$ ($C_2$) x 2, $A_4^2.(A_4^2:C_4\times C_2^2)$ ($C_2$)

Order 165888: $C_2^8.C_3^4.C_2^3$ ($C_4$), $C_2^9.C_3^4:C_4$ ($C_2^2$), $C_2^9.C_3^4:C_4$ ($C_4$)

Order 82944: $C_2^8.C_3^3.D_6$ ($C_8$) x 2, $A_4^2:(A_4^2:C_4)$ ($D_4$) x 2, $C_2^8.C_3^3.D_6$ ($C_2\times C_4$)

Order 41472: $C_2^8.C_3^4.C_2$ ($\OD_{16}$), $C_2^8.C_3^4.C_2$ ($C_2\times C_8$), $C_2^8.C_3^4.C_2$ ($C_2^2:C_4$)

Order 20736: $C_2^8.C_3^4$ ($C_2^2:C_8$)

Order 512: $C_2^9$ ($C_3^4:\OD_{16}$)

Order 256: $C_2^8$ ($(C_3^3\times C_6).\OD_{16}$)

Order 2: $C_2$ ($C_2^8.C_3^4:\OD_{16}$)

Order 1: $C_1$ ($C_2^9.C_3^4:\OD_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

The $146 \times 146$ character table is not available for this group.

Rational character table

The $124 \times 124$ rational character table is not available for this group.