Abstract group 9216.fp: $C_2^5.(A_4\times S_4)$

• Label: 9216.fp
• Order: \( 2^{10} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $20$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_2^5.(A_4\times S_4)$
Order:	\(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^8.C_3.C_6^2.C_2^4$, of order \(442368\)\(\medspace = 2^{14} \cdot 3^{3} \)
Composition factors:	$C_2$ x 10, $C_3$ x 2
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	511	1088	1536	3008	3072	9216
Conjugacy classes	1	43	5	16	31	16	112
Divisions	1	43	3	8	17	4	76
Autjugacy classes	1	17	3	4	11	2	38

Dimension	1	2	3	4	6	9	12	18
Irr. complex chars.	12	6	40	0	20	12	0	22	112
Irr. rational chars.	2	5	8	3	18	6	9	25	76

Minimal presentations

Permutation degree:	$20$
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:	${\langle a, b, c, d, e, f, g, h, i \mid a^{12}=b^{6}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,3)(2,5,10,12)(4,8)(6,11)(7,9)(13,14,16,17,18,15)(19,20), (1,2,4,7,10,11)(3,6,5,9,8,12)(14,15,17)(16,18,19)\rangle$
Transitive group:	36T7713	36T8096			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:	$(C_2^5:A_4)$ . $S_4$ (3)	$C_2^7$ . $(S_3\times A_4)$	$C_2^5$ . $(A_4\times S_4)$ (3)	$(C_2^9.C_3)$ . $S_3$	all 33

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

Homology

Abelianization:	$C_{12} \simeq C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 51 normal subgroups (27 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:	a subgroup isomorphic to 768.1085323
Frattini:	a subgroup isomorphic to $C_2^3$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2$

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

Order 9216: $C_2^5.(A_4\times S_4)$

Order 4608: $C_2^9:C_3^2$

Order 3072: $C_2^6.C_6.C_2^3$, $C_2^4:C_3.C_2^5.C_2$

Order 2304: $(A_4\times C_2^4):C_{12}$ x 3, $C_2^4:A_4^2$, $A_4\times C_2^3.S_4$

Order 1536: $C_2^8.C_6$ x 6, $C_2\times C_2^4.(C_6.D_4)$ x 2, $C_2^9.C_3$ x 2, $C_2^9.C_3$

Order 1152: $C_2^3:A_4^2$ x 7

Order 1024: $C_2^9.C_2$

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

Order 576: $C_2^2:A_4^2$ x 7, $A_4^2:C_4$ x 3, $C_3\times C_2^3.S_4$, $(C_2^3\times C_6):C_{12}$

Order 512: $C_2^7:C_4$ x 14, $C_2^9$

Order 384: $C_2^5:A_4$ x 68, $C_2^5:A_4$ x 35, $C_2^5:C_{12}$ x 12, $A_4\times C_2^5$ x 7, $C_2^5:C_{12}$ x 6, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4.S_4$ x 4

Order 288: $C_2\times A_4^2$ x 14, $C_2^5:C_3^2$ x 2

Order 256: $C_2^6:C_4$ x 88, $C_2^6:C_4$ x 50, $C_2^8$ x 43

Order 192: $C_2^2\wr C_3$ x 68, $A_4\times C_2^4$ x 63, $C_2^4:A_4$ x 35, $C_2^4:C_{12}$ x 12, $C_2.\GL(2,\mathbb{Z}/4)$ x 12, $C_2^3.S_4$ x 6, $C_2^4:C_{12}$ x 4, $C_2^4:C_{12}$ x 4, $C_2^3.S_4$ x 2, $C_2^4:C_{12}$, $C_2^4.D_6$

Order 144: $A_4^2$ x 14, $A_4:C_{12}$ x 3, $C_2^4:C_3^2$ x 2, $A_4\times C_3:C_4$

Order 128: $C_2^7$ x 4

Order 96: $C_2^3:A_4$ x 2

Order 64: $C_2^6$ x 4

Order 48: $C_2^2:A_4$ x 2

Order 32: $C_2^5$ x 5

Order 16: $C_2^4$ x 5

Order 8: $C_2^3$ x 4

Order 4: $C_2^2$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 9216: $C_2^5.(A_4\times S_4)$ ($C_1$)

Order 4608: $C_2^9:C_3^2$ ($C_2$)

Order 3072: $C_2^4:C_3.C_2^5.C_2$ ($C_3$)

Order 2304: $C_2^4:A_4^2$ ($C_4$)

Order 1536: $C_2^9.C_3$ ($C_6$), $C_2^9.C_3$ ($S_3$)

Order 768: $C_2^2\times C_2^2:(A_4:C_4)$ ($A_4$), $C_2^8.C_3$ ($C_{12}$), $C_2^8.C_3$ ($C_3:C_4$)

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

Order 384: $C_2^5:A_4$ ($S_4$) x 3, $C_2^5:A_4$ ($C_2\times A_4$)

Order 256: $C_2^8$ ($C_3:C_{12}$)

Order 192: $C_2^2\wr C_3$ ($A_4:C_4$) x 3, $C_2^2\wr C_3$ ($C_4\times A_4$)

Order 128: $C_2^7$ ($C_3\times S_4$) x 3, $C_2^7$ ($S_3\times A_4$)

Order 96: $C_2^3:A_4$ ($C_2^3:A_4$), $C_2^3:A_4$ ($C_2^2:S_4$)

Order 64: $C_2^6$ ($A_4:C_{12}$) x 3, $C_2^6$ ($A_4\times C_3:C_4$)

Order 48: $C_2^2:A_4$ ($C_2^4:C_{12}$), $C_2^2:A_4$ ($C_2^3.S_4$)

Order 32: $C_2^5$ ($A_4\times S_4$) x 3, $C_2^5$ ($(C_2\times D_6):A_4$), $C_2^5$ ($C_3\times C_2^2:S_4$)

Order 16: $C_2^4$ ($A_4^2:C_4$) x 3, $C_2^4$ ($C_3\times C_2^3.S_4$), $C_2^4$ ($(C_2^3\times C_6):C_{12}$)

Order 8: $C_2^3$ ($(C_2^2\times S_4):A_4$) x 3, $C_2^3$ ($A_4\times C_2^2:S_4$)

Order 4: $C_2^2$ ($(A_4\times C_2^4):C_{12}$) x 3, $C_2^2$ ($A_4\times C_2^3.S_4$)

Order 2: $C_2$ ($C_2^6:(C_3\times S_4)$)

Order 1: $C_1$ ($C_2^5.(A_4\times S_4)$)

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 10 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 $112 \times 112$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $76 \times 76$ rational character table.