Abstract group 21499084800000000.gt: $A_5^8.D_4^2.C_2$

• Label: 214...000.gt
• Order: \( 2^{23} \cdot 3^{8} \cdot 5^{8} \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{25} \cdot 3^{8} \cdot 5^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: not computed
• Trans deg.: $40$
• Rank: $3$

Group information

Description:	$A_5^8.D_4^2.C_2$
Order:	\(21499084800000000\)\(\medspace = 2^{23} \cdot 3^{8} \cdot 5^{8} \)
Exponent:	\(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \)
Automorphism group:	Group of order \(85996339200000000\)\(\medspace = 2^{25} \cdot 3^{8} \cdot 5^{8} \)
Composition factors:	$C_2$ x 7, $A_5$ x 8
Derived length:	$3$

This group is nonabelian and nonsolvable. Whether it is rational has not been computed.

Group statistics

Statistics about orders of elements in this group have not been computed.

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$40$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Permutation group:	Degree $40$ $\langle(1,16,4,17,3,20,2,18,5,19)(6,31,8,34,7,33,10,35,9,32)(11,29)(12,28,13,26,15,30) \!\cdots\! \rangle$
Transitive group:	40T314111				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:	$(A_5^8.D_4:D_4)$ . $C_2$	$(A_5^8.C_2\wr C_4)$ . $C_2$	$(A_5^8.C_2\wr C_4)$ . $C_2$	$(A_5^8.C_4^2:C_4)$ . $C_2$	all 17

Elements of the group are displayed as permutations of degree 40.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 29 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^4.C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^8$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^8$

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 225 (order at least 95551488000000) are shown, as well as all normal subgroups of any index.

Order 21499084800000000: $A_5^8.D_4^2.C_2$

Order 10749542400000000: $A_5^4.A_5^4.D_4^2$, $A_5^8.C_4^2:C_4$, $A_5^8.C_2\wr C_4$, $A_5^8.C_2\wr C_4$, $A_5^8.D_4:D_4$, $A_5^8.C_2\wr C_2^2$, $A_5^8.C_2\wr C_2^2$

Order 5374771200000000: $A_5^4.A_5^4.C_2^2\wr C_2$ x 3, $A_5^8.C_2^3.C_2^2$ x 2, $A_5^4.A_5^4.C_4\wr C_2$, $A_5^4.A_5^4.C_4:D_4$, $A_5^4.A_5^4.C_2^4.C_2$, $A_5^4.A_5^4.C_2^3.C_2^2$, $A_5^4.A_5^4.C_2^2:D_4$, $A_5^4.A_5^4.C_2.C_2^4$, $A_5^2.A_5^2.A_5^4.C_2^2:D_4$, $A_5^2.A_5^2.A_5^4.C_2.C_2^4$, $A_5^2.A_5^2.A_5^4.(C_4\times D_4)$, $A_5^8.C_2^4.C_2$, $A_5^8.D_8.C_2$, $A_5^8.C_2^3.C_2^2$, $A_5^8.D_4.C_2^2$, $A_5^8.(C_2^3.C_4)$

Order 2687385600000000: $A_5^4.A_5^4.C_2^3.C_2$ x 8, $A_5^4.A_5^4.D_4.C_2$ x 6, $A_5^2.A_5^2.A_5^4.C_2^3.C_2$ x 5, $A_5^2.A_5^2.A_5^4.D_4.C_2$ x 3, $A_5^2.A_5^2.A_5^4.C_4.C_2^2$ x 2, $A_5^2.A_5^2.A_5^4.C_2^2:C_4$ x 2, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^3.C_2$ x 2, $A_5^4.A_5^4.\OD_{16}$, $A_5^4.A_5^4.C_2^4$, $A_5^4.A_5^4.C_2^2:C_4$, $A_5^2.A_5^2.A_5^4.C_2^4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_4^2$, $A_5^2.A_5^2.A_5^2.A_5^2.C_4:C_4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2:C_4$, $A_5.A_5.A_5.A_5.A_5^4.C_4.C_2^2$, $A_5^8.D_8$, $A_5^8.C_2^2:C_4$, $A_5^8.\SD_{16}$, $A_5^8.D_4.C_2$, $A_5^8.D_4:C_2$, $A_5^8.D_4.C_2$

Order 1343692800000000: $A_5^4.A_5^4.D_4$ x 9, $A_5^2.A_5^2.A_5^4.D_4$ x 8, $A_5^2.A_5^2.A_5^2.A_5^2.D_4$ x 7, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^3$ x 7, $A_5^4.A_5^4.C_2^3$ x 5, $A_5^2.A_5^2.A_5^4.C_2^3$ x 5, $A_5^4.A_5^4.C_4.C_2$ x 3, $A_5^4.A_5^4.C_2^2.C_2$ x 2, $A_5^2.A_5^2.A_5^4.C_4.C_2$ x 2, $A_5.A_5.A_5.A_5.A_5^4.C_4.C_2$ x 2, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.C_4.C_2$ x 2, $A_5^8.D_4$ x 2, $A_5^4.A_5^4.Q_8$, $A_5^4.A_5^4.C_8$, $A_5^2.A_5^2.A_5^2.A_5^2.C_4.C_2$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2.C_2$, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.C_2^3$

Order 671846400000000: $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2$ x 15, $A_5^4.A_5^4.C_2^2$ x 5, $A_5^2.A_5^2.A_5^4.C_2^2$ x 4, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.C_2^2$ x 4, $A_5^2.A_5^2.A_5^2.A_5^2.C_4$ x 3, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.C_4$ x 3, $A_5^4.A_5^4.C_4$ x 2, $A_5.A_5.A_5.A_5.A_5.A_5.A_5.A_5.C_2^2$

Order 537477120000000: $A_4.A_5.A_5.A_5.A_5^4.C_4.C_2^2$

Order 447897600000000: $D_5.A_5.A_5.A_5.A_5^4.C_4.C_2^2$

Order 335923200000000: $A_5^2.A_5^2.A_5^2.A_5^2.C_2$ x 6, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.C_2$ x 2, $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5$, $A_5.A_5.A_5.A_5.A_5.A_5.A_5.A_5.C_2$

Order 268738560000000: $A_4.A_5.A_5.A_5.A_5^4.C_4.C_2$ x 3, $A_4.A_5.A_5.A_5.A_5^2.A_5^2.C_4.C_2$ x 2, $A_4.A_5.A_5.A_5.A_5^4.C_2^2.C_2$, $A_4.A_5.A_5.A_5.A_5^2.A_5^2.C_2^3$, $(S_3\times A_5).A_5.A_5.A_5^4.C_4.C_2^2$

Order 223948800000000: $D_5.A_5.A_5.A_5.A_5^4.C_4.C_2$ x 3, $D_5.A_5.A_5.A_5.A_5^2.A_5^2.C_4.C_2$ x 2, $D_5.A_5.A_5.A_5.A_5^4.C_2^2.C_2$, $D_5.A_5.A_5.A_5.A_5^2.A_5^2.C_2^3$

Order 214990848000000: $A_4^2.A_5^2.A_5^4.C_4.C_2^3$, $A_4^2.A_5^2.A_5^4.C_2^2:D_4$, $A_4^2.A_5^2.A_5^4.(C_4\times D_4)$

Order 179159040000000: $(C_2^2\times A_5).A_5.A_5.A_5^4.C_4.C_2^2$

Order 167961600000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.A_5$

Order 149299200000000: $D_5^2.A_5^2.A_5^4.C_2^2:D_4$, $D_5^2.A_5^2.A_5^4.C_2.C_2^4$, $D_5^2.A_5^2.A_5^4.(C_4\times D_4)$

Order 134369280000000: $(C_3\times A_5).A_5.A_5.A_5^4.C_4.C_2^2$ x 8, $A_4.A_5.A_5.A_5.A_5^2.A_5^2.C_2^2$ x 6, $A_4.A_5.A_5.A_5.A_5^2.A_5^2.C_4$ x 4, $(S_3\times A_5).A_5.A_5.A_5^4.C_4.C_2$ x 3, $(S_3\times A_5).A_5.A_5.A_5^2.A_5^2.C_4.C_2$ x 2, $A_4.A_5.A_5.A_5.A_5.A_5.A_5.A_5.C_2^2$, $(S_3\times A_5).A_5.A_5.A_5^4.C_2^2.C_2$, $(S_3\times A_5).A_5.A_5.A_5^2.A_5^2.C_2^3$

Order 111974400000000: $(C_5\times A_5).A_5.A_5.A_5^4.C_4.C_2$ x 8, $D_5.A_5.A_5.A_5.A_5^2.A_5^2.C_2^2$ x 6, $D_5.A_5.A_5.A_5.A_5^2.A_5^2.C_4$ x 4, $D_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2$

Order 107495424000000: $A_4^2.A_5^2.A_5^4.D_4.C_2$ x 7, $A_4^2.A_5^2.A_5^4.C_2^3.C_2$ x 4, $A_4^2.A_5^2.A_5^4.C_4.C_2^2$ x 3, $A_4^2.A_5.A_5.A_5^4.C_4.C_2^2$ x 3, $A_4^2.A_5^2.A_5^4.C_2^2:C_4$ x 2, $A_4^2.A_5^2.A_5^2.A_5^2.D_4.C_2$ x 2, $A_4^2.A_5^2.A_5^2.A_5^2.C_4:C_4$ x 2, $A_4^2.A_5^2.A_5^4.C_2^4$, $A_4^2.A_5^2.A_5^2.A_5^2.C_4^2$, $A_4^2.A_5^2.A_5^2.A_5^2.C_4.C_2^2$, $A_4^2.A_5^2.A_5^2.A_5^2.C_2^4$, $A_4^2.A_5^2.A_5^2.A_5^2.C_2^3.C_2$, $A_4^2.A_5^2.A_5^2.A_5^2.C_2^2:C_4$

Order 55037657088: $C_2^{16}.C_3^4.C_3^4.D_4^2.C_2$

Order 12800000000: $C_5^8.C_2^4.C_2^6.C_2^4.C_2$

Order 1658880000: $A_5^4.D_4^2.C_2$ x 2

Order 214990848: $C_3^8.C_2^6.C_2^5.C_2^4$

Order 8388608: $C_2^9.C_2^6.C_2^4.C_2^4$

Order 390625: $C_5^8$

Order 6561: $C_3^8$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 21499084800000000: $A_5^8.D_4^2.C_2$ ($C_1$)

Order 10749542400000000: $A_5^4.A_5^4.D_4^2$ ($C_2$), $A_5^8.C_4^2:C_4$ ($C_2$), $A_5^8.C_2\wr C_4$ ($C_2$), $A_5^8.C_2\wr C_4$ ($C_2$), $A_5^8.D_4:D_4$ ($C_2$), $A_5^8.C_2\wr C_2^2$ ($C_2$), $A_5^8.C_2\wr C_2^2$ ($C_2$)

Order 5374771200000000: $A_5^4.A_5^4.C_2^2\wr C_2$ ($C_2^2$) x 2, $A_5^8.C_2^3.C_2^2$ ($C_2^2$) x 2, $A_5^4.A_5^4.C_4:D_4$ ($C_2^2$), $A_5^8.D_4.C_2^2$ ($C_2^2$), $A_5^8.(C_2^3.C_4)$ ($C_2^2$)

Order 2687385600000000: $A_5^4.A_5^4.C_2^4$ ($D_4$), $A_5^4.A_5^4.C_2^3.C_2$ ($C_2^3$), $A_5^4.A_5^4.C_2^3.C_2$ ($D_4$), $A_5^4.A_5^4.C_2^2:C_4$ ($D_4$), $A_5^2.A_5^2.A_5^2.A_5^2.C_4^2$ ($D_4$), $A_5^2.A_5^2.A_5^2.A_5^2.C_2^4$ ($D_4$), $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2:C_4$ ($D_4$)

Order 1343692800000000: $A_5^2.A_5^2.A_5^2.A_5^2.C_2^3$ ($C_2\times D_4$) x 2, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2.C_2$ ($C_2\times D_4$)

Order 671846400000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.A_5.C_2^2$ ($C_2^2\wr C_2$)

Order 335923200000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5$ ($C_2\wr C_2^2$)

Order 167961600000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.A_5$ ($C_2\wr D_4$)

Order 1: $C_1$ ($A_5^8.D_4^2.C_2$)

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 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

The character tables for this group have not been computed.