Abstract group 85996339200000000.gj: $A_5^8.C_4.C_4^2.D_4$

• Label: 859...000.gj
• Order: \( 2^{25} \cdot 3^{8} \cdot 5^{8} \)
• Exponent: \( 2^{5} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{27} \cdot 3^{8} \cdot 5^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: not computed
• Trans deg.: $40$
• Rank: $3$

Group information

Description:	$A_5^8.C_4.C_4^2.D_4$
Order:	\(85996339200000000\)\(\medspace = 2^{25} \cdot 3^{8} \cdot 5^{8} \)
Exponent:	\(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Automorphism group:	Group of order \(343985356800000000\)\(\medspace = 2^{27} \cdot 3^{8} \cdot 5^{8} \)
Composition factors:	$C_2$ x 9, $A_5$ x 8
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	16	20	24	30	32	40	48	60	80	120	240
Elements	1	12589527295	37822859360	87722844000000	152587890624	180307158480800	2197785415680000	142262122582080	3788527503360000	16624714640640	11674613514240000	1727571456000000	14828011683840000	2632079904019200	10749542400000000	2493734584320000	15098196787200000	11160189296640000	352266485760000	7774228316160000	1092472012800000	85996339200000000
Conjugacy classes	1	42	20	228	30	696	144	366	2029	273	54	603	601	2328	4	156	96	2943	42	432	62	11150
Divisions	1	42	20	222	30	696	128	366	1972	273	47	594	509	2328	2	118	78	2878	28	308	39	10679
Autjugacy classes	1	42	20	217	27	684	109	342	1895	252	34	573	458	2171	1	111	67	2710	28	301	39	10082

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 $40$ $\langle(1,40,13,6,23,17,35,30,4,36,15,8,21,20,32,28,2,37,11,9,25,16,31,29,3,38,12,10,24,19,34,26) \!\cdots\! \rangle$
Transitive group:	40T314811				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:	$(A_5^8.C_4.C_4^2)$ . $D_4$ (2)	$(A_5^8.C_4.C_4^2)$ . $D_4$ (2)	$(A_5^8.C_{16}.C_4.C_4)$ . $C_2$ (2)	$(A_5^8.\OD_{32}.C_2^2)$ . $C_2^2$ (2)	all 29

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

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 61 normal subgroups (39 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 $C_1$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^4.C_2^5.C_2$
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 75 (order at least 1146617856000000) are shown, as well as all normal subgroups of any index.

Order 85996339200000000: $A_5^8.C_4.C_4^2.D_4$

Order 42998169600000000: $A_5^8.C_{16}.C_4.C_4$ x 2, $A_5^8.C_4.C_4^2.C_2^2$ x 2, $A_5^4.A_5^4.C_4^2.C_2^4$, $A_5^8.C_4.C_4^2.C_2^2$, $A_5^8.C_4^2.C_2^3.C_2$

Order 21499084800000000: $A_5^4.A_5^4.C_4^2.C_2^3$ x 6, $A_5^4.A_5^4.C_2.C_8:D_4$ x 4, $A_5^8.C_4.C_4^2.C_2$ x 2, $A_5^8.C_4.C_4^2.C_2$ x 2, $A_5^8.C_4.C_4^2.C_2$ x 2, $A_5^8.C_4.C_4^2.C_2$ x 2, $A_5^8.\OD_{32}.C_2^2$ x 2, $A_5^8.C_4.C_4^2.C_2$ x 2, $A_5^4.A_5^4.C_2^2.C_2^4.C_2$, $A_5^2.A_5^2.A_5^4.C_4.C_2^4.C_2$, $A_5^8.(C_8.C_8).C_2$, $A_5^8.C_4:\SD_{16}.C_2$, $A_5^8.\OD_{16}:C_2^2.C_2$

Order 10749542400000000: $A_5^4.A_5^4.D_8:C_4$ x 4, $A_5^4.A_5^4.C_4:D_4.C_2$ x 4, $A_5^4.A_5^4.C_2^2:D_4.C_2$ x 3, $A_5^4.A_5^4.C_8:Q_8$ x 2, $A_5^4.A_5^4.C_8:D_4$ x 2, $A_5^4.A_5^4.C_8:C_8$ x 2, $A_5^4.A_5^4.C_4:\SD_{16}$ x 2, $A_5^4.A_5^4.C_4:Q_8.C_2$ x 2, $A_5^4.A_5^4.C_4.C_4^2$ x 2, $A_5^4.A_5^4.C_4.C_2^4$ x 2, $A_5^4.A_5^4.C_4.C_2^3.C_2$ x 2, $A_5^2.A_5^2.A_5^4.C_8:D_4$ x 2, $A_5^2.A_5^2.A_5^4.C_4:\SD_{16}$ x 2, $A_5^2.A_5^2.A_5^4.C_2.C_4:D_4$ x 2, $A_5^8.C_2^2.C_4^2$ x 2, $A_5^8.C_{16}.C_4$ x 2, $A_5^8.(C_8.C_8)$ x 2, $A_5^8.C_4.C_4^2$ x 2, $A_5^8.C_4.C_4^2$ x 2, $A_5^8.\OD_{16}:C_2^2$ x 2, $A_5^8.\OD_{32}.C_2$ x 2, $A_5^4.A_5^4.C_8^2$, $A_5^4.A_5^4.C_4^2.C_2^2$, $A_5^4.A_5^4.C_4.D_4:C_2$, $A_5^4.A_5^4.C_2^4.C_2^2$, $A_5^4.A_5^4.C_2^2.C_2^4$, $A_5^4.A_5^4.C_2.C_2^4.C_2$, $A_5^2.A_5^2.A_5^4.C_4.C_2^4$, $A_5^2.A_5^2.A_5^4.C_2^2:D_4.C_2$, $A_5^2.A_5^2.A_5^4.C_2^2.C_2^4$, $A_5^2.A_5^2.A_5^4.(C_4\times \OD_{16})$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^3.C_2^3$, $A_5.A_5.A_5^2.A_5^4.C_8.C_2^3$, $A_5^8.C_2^3.C_2^3$, $A_5^8.C_4.C_2^4$

Order 5374771200000000: $A_5^4.A_5^4.C_4.C_2^3$ x 19, $A_5^2.A_5^2.A_5^4.C_4.C_2^3$ x 9, $A_5^4.A_5^4.D_4:C_4$ x 6, $A_5^4.A_5^4.C_4^2.C_2$ x 5, $A_5^2.A_5^2.A_5^4.C_2.C_4^2$ x 4, $A_5^2.A_5^2.A_5^2.A_5^2.C_4:D_4$ x 4, $A_5.A_5.A_5^2.A_5^4.D_8.C_2$ x 4, $A_5^4.A_5^4.C_8:C_4$ x 3, $A_5^4.A_5^4.C_2^3.C_2^2$ x 3, $A_5^4.A_5^4.(C_4\times D_4)$ x 3, $A_5.A_5.A_5^2.A_5^4.C_4.C_2^3$ x 3, $A_5^2.A_5^2.A_5^4.D_4:C_4$ x 2, $A_5^2.A_5^2.A_5^4.C_2^2:D_4$ x 2, $A_5^2.A_5^2.A_5^4.(C_4\times D_4)$ x 2, $A_5.A_5.A_5^2.A_5^4.\SD_{16}.C_2$ x 2, $A_5^8.C_4\wr C_2$ x 2, $A_5^8.C_4.C_2^3$ x 2, $A_5^8.\OD_{32}$ x 2, $A_5^4.A_5^4.C_4:Q_8$, $A_5^4.A_5^4.C_4:D_4$, $A_5^4.A_5^4.C_2.C_2^4$, $A_5^4.A_5^4.(C_2^3.C_4)$, $A_5^2.A_5^2.A_5^4.C_8:C_4$, $A_5^2.A_5^2.A_5^4.C_8.C_2^2$, $A_5^2.A_5^2.A_5^4.C_4:Q_8$, $A_5^2.A_5^2.A_5^4.C_4:D_4$, $A_5^2.A_5^2.A_5^4.C_2^4.C_2$, $A_5^2.A_5^2.A_5^4.(C_4.D_4)$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^4.C_2$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2:D_4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2.C_2^3$, $A_5.A_5.A_5^2.A_5^2.A_5^2.C_4.C_2^3$, $A_5.A_5.A_5.A_5.A_5^4.\OD_{16}.C_2$, $A_5^8.\OD_{16}.C_2$

Order 2687385600000000: $A_5^2.A_5^2.A_5^2.A_5^2.C_2^3.C_2$ x 12, $A_5^4.A_5^4.D_8$ x 8, $A_5^4.A_5^4.\OD_{16}$ x 6, $A_5^4.A_5^4.C_4.C_2^2$ x 6, $A_5^2.A_5^2.A_5^4.D_8$ x 6, $A_5.A_5.A_5^2.A_5^2.A_5^2.D_4.C_2$ x 6, $A_5^4.A_5^4.\SD_{16}$ x 4, $A_5^4.A_5^4.D_4:C_2$ x 4, $A_5^2.A_5^2.A_5^4.\SD_{16}$ x 4, $A_5.A_5.A_5^2.A_5^4.\SD_{16}$ x 4, $A_5.A_5.A_5^2.A_5^4.D_8$ x 4, $A_5^2.A_5^2.A_5^4.C_4.C_2^2$ x 3, $A_5^2.A_5^2.A_5^4.C_2^3.C_2$ x 3, $A_5^2.A_5^2.A_5^4.C_2^2:C_4$ x 3, $A_5^2.A_5^2.A_5^2.A_5^2.C_4^2$ x 3, $A_5^4.A_5^4.C_4^2$ x 2, $A_5^4.A_5^4.C_4:C_4$ x 2, $A_5^4.A_5^4.C_2.C_2^3$ x 2, $A_5^2.A_5^2.A_5^4.\OD_{16}$ x 2, $A_5^2.A_5^2.A_5^4.C_4:C_4$ x 2, $A_5.A_5.A_5^2.A_5^4.Q_8.C_2$ x 2, $A_5.A_5.A_5^2.A_5^4.D_4:C_2$ x 2, $A_5.A_5.A_5^2.A_5^4.D_4.C_2$ x 2, $A_5.A_5.A_5.A_5.A_5^4.\OD_{16}$ x 2, $A_5.A_5.A_5.A_5.A_5^4.C_8.C_2$ x 2, $A_5^8.C_8.C_2$ x 2, $A_5^8.\OD_{16}$ x 2, $A_5^8.C_{16}$ x 2, $A_5^4.A_5^4.C_2^3.C_2$, $A_5^2.A_5^2.A_5^4.D_4.C_2$, $A_5^2.A_5^2.A_5^2.A_5^2.C_4:C_4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2:C_4$, $A_5.A_5.A_5^2.A_5^4.C_4.C_2^2$, $A_5.A_5.A_5^2.A_5^2.A_5^2.C_2^3.C_2$, $A_5.A_5.A_5^2.A_5^2.A_5^2.C_2.C_2^3$, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.C_4.C_2^2$, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.C_2^4$

Order 2149908480000000: $A_4.A_5.A_5^2.A_5^4.D_4.C_2^3$

Order 1791590400000000: $D_5.A_5.A_5^2.A_5^4.C_8.C_2^3$

Order 1343692800000000: $A_5^2.A_5^2.A_5^2.A_5^2.D_4$ x 10, $A_5.A_5.A_5^2.A_5^2.A_5^2.D_4$ x 8, $A_5^4.A_5^4.C_8$ x 6, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2.C_2$ x 6, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.C_2^3$ x 6, $A_5^2.A_5^2.A_5^4.D_4$ x 4, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.C_4.C_2$ x 4, $A_5^2.A_5^2.A_5^4.C_8$ x 3, $A_5.A_5.A_5.A_5.A_5^4.C_8$ x 3, $A_5^4.A_5^4.Q_8$ x 2, $A_5^4.A_5^4.D_4$ x 2, $A_5^4.A_5^4.C_4.C_2$ x 2, $A_5.A_5.A_5^2.A_5^4.Q_8$ x 2, $A_5.A_5.A_5^2.A_5^4.D_4$ x 2, $A_5^2.A_5^2.A_5^4.C_4.C_2$, $A_5^2.A_5^2.A_5^4.C_2^2.C_2$, $A_5^2.A_5^2.A_5^2.A_5^2.C_4.C_2$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^3$, $A_5.A_5.A_5^2.A_5^4.C_4.C_2$, $A_5.A_5.A_5^2.A_5^4.C_2^2.C_2$, $A_5.A_5.A_5^2.A_5^2.A_5^2.C_4.C_2$, $A_5.A_5.A_5^2.A_5^2.A_5^2.C_2^3$, $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2^2$

Order 671846400000000: $A_5^2.A_5^2.A_5^2.A_5^2.C_4$ x 2, $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2$

Order 335923200000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5$

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

Order 220150628352: $C_2^{16}.C_3^8.C_4^2.C_2^4.C_2$

Order 51200000000: $D_5^4.(D_5^4.C_8).\OD_{16}.C_2^2$

Order 859963392: $C_3^8.C_2^6.C_2^4.C_2^5.C_2^2$

Order 33554432: $C_2^9.C_2^6.C_2^4.C_2^5.C_2$

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 85996339200000000: $A_5^8.C_4.C_4^2.D_4$ ($C_1$)

Order 42998169600000000: $A_5^8.C_{16}.C_4.C_4$ ($C_2$) x 2, $A_5^8.C_4.C_4^2.C_2^2$ ($C_2$) x 2, $A_5^4.A_5^4.C_4^2.C_2^4$ ($C_2$), $A_5^8.C_4.C_4^2.C_2^2$ ($C_2$), $A_5^8.C_4^2.C_2^3.C_2$ ($C_2$)

Order 21499084800000000: $A_5^4.A_5^4.C_4^2.C_2^3$ ($C_2^2$) x 3, $A_5^4.A_5^4.C_4^2.C_2^3$ ($C_4$) x 2, $A_5^4.A_5^4.C_2.C_8:D_4$ ($C_4$) x 2, $A_5^8.C_4.C_4^2.C_2$ ($C_2^2$) x 2, $A_5^8.\OD_{32}.C_2^2$ ($C_2^2$) x 2

Order 10749542400000000: $A_5^4.A_5^4.C_4:D_4.C_2$ ($C_2\times C_4$) x 2, $A_5^8.C_4.C_4^2$ ($D_4$) x 2, $A_5^8.C_4.C_4^2$ ($D_4$) x 2, $A_5^4.A_5^4.C_8^2$ ($C_2\times C_4$), $A_5^4.A_5^4.C_8:Q_8$ ($C_2\times C_4$), $A_5^4.A_5^4.C_8:C_8$ ($C_2\times C_4$), $A_5^4.A_5^4.C_4^2.C_2^2$ ($C_2^3$), $A_5^4.A_5^4.C_4:\SD_{16}$ ($D_4$), $A_5^4.A_5^4.C_4:\SD_{16}$ ($C_2\times C_4$), $A_5^4.A_5^4.C_4:D_4.C_2$ ($D_4$), $A_5^4.A_5^4.C_2^2.C_2^4$ ($D_4$), $A_5^2.A_5^2.A_5^2.A_5^2.C_2^3.C_2^3$ ($D_4$)

Order 5374771200000000: $A_5^4.A_5^4.C_4^2.C_2$ ($D_4:C_2$) x 2, $A_5^4.A_5^4.C_4.C_2^3$ ($C_2\times D_4$) x 2, $A_5^2.A_5^2.A_5^2.A_5^2.C_4:D_4$ ($C_2^2:C_4$) x 2, $A_5^4.A_5^4.C_4^2.C_2$ ($C_2\times D_4$), $A_5^4.A_5^4.C_4^2.C_2$ ($C_2^2\times C_4$), $A_5^4.A_5^4.C_4:Q_8$ ($C_2^2:C_4$), $A_5^4.A_5^4.C_4:D_4$ ($C_2^2:C_4$), $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2.C_2^3$ ($C_2\times D_4$)

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

Order 1343692800000000: $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2.C_2$ ($C_2^3.D_4$), $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2.C_2$ ($D_4:D_4$), $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2^2$ ($D_4:D_4$)

Order 671846400000000: $A_5^2.A_5^2.A_5^2.A_5^2.C_4$ ($C_4^2.D_4$) x 2, $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2$ ($C_2^4.D_4$)

Order 335923200000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5$ ($C_4^2.(C_2\times D_4)$)

Order 167961600000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.A_5$ ($C_8^2:(C_2\times C_4)$)

Order 1: $C_1$ ($A_5^8.C_4.C_4^2.D_4$)

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 3 larger groups in the database.

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

Character theory

Complex character table

The $11150 \times 11150$ character table is not available for this group.

Rational character table

The $10679 \times 10679$ rational character table is not available for this group.