Abstract group 257989017600000000.q: $A_5^8.C_4^3.S_4$

• Label: 257...000.q
• Order: \( 2^{25} \cdot 3^{9} \cdot 5^{8} \)
• Exponent: \( 2^{4} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{27} \cdot 3^{9} \cdot 5^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: not computed
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$A_5^8.C_4^3.S_4$
Order:	\(257989017600000000\)\(\medspace = 2^{25} \cdot 3^{9} \cdot 5^{8} \)
Exponent:	\(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \)
Automorphism group:	Group of order \(1031956070400000000\)\(\medspace = 2^{27} \cdot 3^{9} \cdot 5^{8} \)
Composition factors:	$C_2$ x 9, $C_3$, $A_5$ x 8
Derived length:	$4$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	15	16	18	20	24	30	36	40	45	48	60	72	80	90	120	240
Elements	1	13994967295	769388939360	161307143520000	152587890624	1041383290320800	6301728460800000	321889075200000	509820717142080	22842684549120000	2115413148560640	20919803904000000	3108409344000000	9067956756480000	60305078845440000	12504072572179200	17915904000000000	10039276339200000	4381035724800000	21164654592000000	24754403450880000	7166361600000000	4013162496000000	5326995456000000	17720341954560000	6306398208000000	257989017600000000
Conjugacy classes	1	28	18	138	22	351	132	7	183	1166	161	23	24	280	612	966	21	137	25	42	1233	2	24	20	393	24	6033
Divisions	1	28	18	132	22	351	103	7	183	1044	151	14	24	267	417	955	19	88	20	22	1123	1	10	18	241	11	5270
Autjugacy classes	1	28	18	130	17	350	99	7	163	1037	134	12	24	241	410	896	19	87	17	21	1069	1	10	17	241	11	5060

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$40$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Permutation group:	Degree $40$ $\langle(1,22,17,30,8,13,5,21,19,29,10,15,2,24,18,27,9,12,3,25,16,28,6,11)(4,23,20,26,7,14) \!\cdots\! \rangle$
Transitive group:	40T315210				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:	$(A_5^8.C_2.C_2^5)$ . $S_4$ (2)	$(A_5^8.C_4^3.A_4)$ . $C_2$	$(A_5^8.C_4^3.C_2^2)$ . $S_3$	$(A_5^2.A_5^2.A_5^2.A_5^2.C_4^3)$ . $S_4$	all 9

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

Homology

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

Subgroups

There are 12 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:	a subgroup isomorphic to $A_5^8.C_4^3.A_4$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^6.C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^8.C_3$
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 180 (order at least 1433272320000000) are shown, as well as all normal subgroups of any index.

Order 257989017600000000: $A_5^8.C_4^3.S_4$

Order 128994508800000000: $A_5^8.C_4^3.A_4$

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

Order 64497254400000000: $A_5^2.A_5^6.C_4^2.D_{12}$, $A_5^8.C_4.C_2^2.S_4$, $A_5^8.D_4:C_2.S_4$

Order 42998169600000000: $A_5^4.A_5^4.C_4^2.C_2.C_2^3$, $A_5^8.C_4^3.C_2^2$, $A_5^8.C_4^2.C_4^2$, $A_5^8.C_4:D_4.C_2^3$, $A_5^8.C_4:Q_8.C_2^3$, $A_5^8.C_4^2.C_2^2.C_2^2$, $A_5^8.C_2\wr C_4.C_2^2$

Order 32248627200000000: $A_5^8.D_4:C_2:A_4$ x 2, $A_5^4.A_5^4.Q_8.S_4$, $A_5^4.A_5^4.C_2^3.S_4$, $A_5^2.A_5^6.C_4^2.D_6$, $A_5^2.A_5^6.C_4^2.C_{12}$, $A_5.A_5.A_5^6.C_4^2.D_6$, $A_5^8.Q_8.S_4$, $A_5^8.C_2^3.S_4$, $A_5^8.C_4.C_4^2.C_3$

Order 21499084800000000: $A_5^4.A_5^4.C_4^2.C_2^3$ x 2, $A_5^8.C_2.C_2\wr C_4$ x 2, $A_5^8.C_2.C_2\wr C_4$ x 2, $A_5^8.C_4:Q_8.C_2^2$ x 2, $A_5^8.D_4^2.C_2$ x 2, $A_5^8.D_4:D_4.C_2$ x 2, $A_5^4.A_5^4.C_4^3.C_2$, $A_5^4.A_5^4.C_2^2.C_2^5$, $A_5^2.A_5^2.A_5^4.C_4.(C_4\times D_4)$, $A_5^8.C_2.C_2\wr C_2^2$, $A_5^8.C_2\wr C_4.C_2$, $A_5^8.C_2\wr C_4.C_2$, $A_5^8.C_2.C_2\wr C_2^2$, $A_5^8.C_4.C_4^2.C_2$, $A_5^8.C_2^2.C_2^4.C_2$, $A_5^8.C_2^2.C_2^4.C_2$, $A_5^8.C_4^2:C_4.C_2$, $A_5^8.C_2\wr C_4.C_2$, $A_5^8.C_4\wr C_2.C_2^2$, $A_5^8.C_2^3.C_2^4$

Order 16124313600000000: $A_5^4.A_5^4.C_2^3.A_4$ x 2, $A_5^8.A_4.D_4$ x 2, $A_5^8.Q_8.D_6$ x 2, $A_5^2.A_5^6.C_4^2.S_3$, $A_5^2.A_5^6.A_4.D_4$, $A_5.A_5.A_5^6.C_4^2.S_3$, $A_5.A_5.A_5^6.C_4^2.C_6$

Order 10749542400000000: $A_5^4.A_5^4.C_2^2.C_2^4$ x 5, $A_5^8.C_4^2.C_2^2$ x 4, $A_5^4.A_5^4.C_2^3.C_2^3$ x 3, $A_5^8.\OD_{16}.C_2.C_2$ x 3, $A_5^4.A_5^4.D_4^2$ x 2, $A_5^4.A_5^4.C_4:C_8.C_2$ x 2, $A_5^4.A_5^4.C_4.C_2.C_2^3$ x 2, $A_5^4.A_5^4.(C_4.D_8)$ x 2, $A_5^8.C_2.C_2^5$ x 2, $A_5^8.C_2.C_2^4.C_2$ x 2, $A_5^8.C_4^2:C_4$ x 2, $A_5^8.C_2\wr C_4$ x 2, $A_5^8.\OD_{16}.C_2.C_2$ x 2, $A_5^8.C_2\wr C_4$ x 2, $A_5^8.C_4:Q_8.C_2$ x 2, $A_5^8.D_4:D_4$ x 2, $A_5^8.(D_4.D_4)$ x 2, $A_5^8.C_4\wr C_2:C_2$ x 2, $A_5^8.C_2\wr C_2^2$ x 2, $A_5^8.C_2\wr C_2^2$ x 2, $A_5^8.C_4\wr C_2.C_2$ x 2, $A_5^4.A_5^4.D_4:D_4$, $A_5^4.A_5^4.C_4:\OD_{16}$, $A_5^4.A_5^4.C_4:D_4.C_2$, $A_5^4.A_5^4.C_2\wr C_2^2$, $A_5^2.A_5^2.A_5^4.C_4\wr C_2.C_2$, $A_5^2.A_5^2.A_5^4.C_4:\OD_{16}$, $A_5^2.A_5^2.A_5^4.C_4:D_4.C_2$, $A_5^2.A_5^2.A_5^4.C_2^2.C_2^4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_4^3$, $A_5.A_5.A_5^2.A_5^4.C_4\wr C_2.C_2$, $A_5^8.C_2\wr C_2^2$, $A_5^8.C_2^3.C_2^3$, $A_5^8.C_2.C_2^4.C_2$, $A_5^8.(C_4.D_4).C_2$, $A_5^8.C_4\wr C_2.C_2$, $A_5^8.D_4.C_2^3$, $A_5^8.C_2^3.C_2^3$, $A_5^8.C_4^2.C_2^2$, $A_5^8.C_4.C_4^2$, $A_5^8.C_4.C_2^4$, $A_5^8.D_4:D_4$

Order 8062156800000000: $A_5^8.Q_8.C_6$ x 3, $A_5^8.A_4.C_4$ x 3, $A_5^4.A_5^4.S_4.C_2$ x 2, $A_5^4.A_5^4.Q_8.S_3$ x 2, $A_5^8.Q_8.S_3$ x 2, $A_5^8.S_4.C_2$ x 2, $A_5^2.A_5^6.S_4.C_2$, $A_5^2.A_5^6.A_4.C_4$, $A_5.A_5.A_5^6.C_4^2.C_3$, $A_5.A_5.A_5^3.A_5^3.S_4.C_2$

Order 6449725440000000: $A_4.A_5.A_5^6.C_4^2.D_6$

Order 5374771200000000: $A_5^4.A_5^4.C_4\wr C_2$ x 6, $A_5^4.A_5^4.C_2^4.C_2$ x 6, $A_5^4.A_5^4.C_2.C_2^4$ x 5, $A_5^4.A_5^4.(C_4.D_4)$ x 5, $A_5^4.A_5^4.C_4^2.C_2$ x 4, $A_5^4.A_5^4.C_2^2:D_4$ x 4, $A_5^2.A_5^2.A_5^4.C_2^2:D_4$ x 4, $A_5^8.D_8.C_2$ x 4, $A_5^8.\OD_{16}.C_2$ x 4, $A_5^4.A_5^4.C_4.C_2^3$ x 3, $A_5^4.A_5^4.C_2^3.C_2^2$ x 3, $A_5^2.A_5^2.A_5^4.(C_4.D_4)$ x 3, $A_5^4.A_5^4.C_4:D_4$ x 2, $A_5^4.A_5^4.C_2^2\wr C_2$ x 2, $A_5^4.A_5^4.C_2.D_4.C_2$ x 2, $A_5^2.A_5^2.A_5^4.C_4^2.C_2$ x 2, $A_5^2.A_5^2.A_5^4.C_4\wr C_2$ x 2, $A_5^2.A_5^2.A_5^4.C_4:D_4$ x 2, $A_5^2.A_5^2.A_5^4.C_4:C_8$ x 2, $A_5^2.A_5^2.A_5^4.C_2.D_4.C_2$ x 2, $A_5^2.A_5^2.A_5^4.(C_4\times D_4)$ x 2, $A_5.A_5.A_5^2.A_5^4.C_4\wr C_2$ x 2, $A_5^8.C_2^4.C_2$ x 2, $A_5^8.C_4\wr C_2$ x 2, $A_5^8.C_2^3.C_2^2$ x 2, $A_5^8.Q_{16}.C_2$ x 2, $A_5^8.D_4.C_2^2$ x 2, $A_5^8.D_4.C_2^2$ x 2, $A_5^8.C_2.C_2^4$ x 2, $A_5^8.(C_2^3.C_4)$ x 2, $A_5^8.C_2^3.C_2^2$ x 2, $A_5^8.C_2^2:D_4$ x 2, $A_5^8.D_8.C_2$ x 2, $D_5.A_5.A_5^6.C_4^2.D_6$, $A_5^4.A_5^4.C_4:Q_8$, $A_5^4.A_5^4.C_4:C_8$, $A_5^4.A_5^4.C_4:C_4.C_2$, $A_5^4.A_5^4.(C_4\times D_4)$, $A_5^4.A_5^4.(C_2^3.C_4)$, $A_5^2.A_5^2.A_5^4.C_4:Q_8$, $A_5^2.A_5^2.A_5^4.C_4.C_2^3$, $A_5^2.A_5^2.A_5^4.C_2^2\wr C_2$, $A_5^2.A_5^2.A_5^4.C_2.C_2^4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_4:D_4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2:D_4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2.C_2^3$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2.D_4.C_2$, $A_5.A_5.A_5^2.A_5^4.\OD_{16}.C_2$, $A_5.A_5.A_5^2.A_5^2.A_5^2.C_4^2.C_2$, $A_5.A_5.A_5.A_5.A_5^4.C_4.C_2^3$, $A_5^8.D_8:C_2$, $A_5^8.C_4^2.C_2$, $A_5^8.\OD_{16}.C_2$, $A_5^8.D_8.C_2$, $A_5^8.C_4^2.C_2$, $A_5^8.C_4:D_4$, $A_5^8.D_4.C_2^2$

Order 4031078400000000: $A_5^4.A_5^4.Q_8.C_3$ x 3, $A_5^4.A_5^4.A_4.C_2$ x 3, $A_5^4.A_5^4.S_4$ x 2, $A_5^8.S_4$ x 2, $A_5^2.A_5^6.S_4$, $A_5^2.A_5^6.D_{12}$, $A_5.A_5.A_5^3.A_5^3.S_4$, $A_5.A_5.A_5^3.A_5^3.A_4.C_2$

Order 3224862720000000: $A_4.A_5.A_5^6.C_4^2.S_3$ x 2, $A_4.A_5.A_5^6.C_4^2.C_6$, $(S_3\times A_5).A_5^6.C_4^2.D_6$

Order 2687385600000000: $A_5^4.A_5^4.D_4.C_2$ x 15, $A_5^4.A_5^4.C_2^3.C_2$ x 12, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^3.C_2$ x 9, $A_5^2.A_5^2.A_5^4.C_2^3.C_2$ x 6, $A_5^4.A_5^4.C_2.C_2^3$ x 5, $A_5^2.A_5^2.A_5^4.C_2^2:C_4$ x 5, $A_5^8.D_4:C_2$ x 5, $A_5^4.A_5^4.\OD_{16}$ x 4, $A_5^2.A_5^2.A_5^4.D_4.C_2$ x 4, $A_5^2.A_5^2.A_5^2.A_5^2.C_4^2$ x 4, $A_5^8.\SD_{16}$ x 4, $A_5^8.D_8$ x 3, $A_5^8.C_4.C_2^2$ x 3, $D_5.A_5.A_5^6.C_4^2.S_3$ x 2, $A_5^4.A_5^4.\SD_{16}$ x 2, $A_5^4.A_5^4.D_8$ x 2, $A_5^2.A_5^2.A_5^4.C_4:C_4$ x 2, $A_5^2.A_5^2.A_5^4.C_4.C_2^2$ x 2, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^2:C_4$ x 2, $A_5.A_5.A_5^2.A_5^4.\OD_{16}$ x 2, $A_5.A_5.A_5^2.A_5^2.A_5^2.C_4^2$ x 2, $A_5.A_5.A_5.A_5.A_5^4.D_4.C_2$ x 2, $A_5^8.C_2^2:C_4$ x 2, $A_5^8.D_4.C_2$ x 2, $A_5^8.Q_8.C_2$ x 2, $A_5^8.D_4.C_2$ x 2, $D_5.A_5.A_5^6.C_4^2.C_6$, $A_5^4.A_5^4.C_4^2$, $A_5^4.A_5^4.C_4.C_2^2$, $A_5^4.A_5^4.C_2^4$, $A_5^2.A_5^2.A_5^4.Q_8.C_2$, $A_5^2.A_5^2.A_5^4.D_4:C_2$, $A_5^2.A_5^2.A_5^4.C_4^2$, $A_5^2.A_5^2.A_5^4.C_2^4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_4:C_4$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^4$, $A_5.A_5.A_5^2.A_5^4.C_8.C_2$, $A_5.A_5.A_5.A_5.A_5^4.Q_8.C_2$, $A_5.A_5.A_5.A_5.A_5^4.D_4:C_2$, $A_5.A_5.A_5.A_5.A_5^4.C_4.C_2^2$, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.D_4.C_2$, $A_5.A_5.A_5.A_5.A_5^2.A_5^2.C_4.C_2^2$, $A_5^8.Q_{16}$, $A_5^8.C_8.C_2$, $A_5^8.C_4^2$, $A_5^8.C_4:C_4$, $A_5^8.\OD_{16}$

Order 2579890176000000: $A_4^2.A_5^6.C_4^2.D_{12}$

Order 2149908480000000: $A_4.A_5.A_5^2.A_5^4.C_4\wr C_2.C_2$, $(C_2^2\times A_5).A_5^6.C_4^2.D_6$

Order 2015539200000000: $A_5^4.A_5^4.A_4$ x 3, $A_5^2.A_5^6.D_6$, $A_5^2.A_5^6.C_{12}$, $A_5.A_5.A_5^3.A_5^3.D_6$, $A_5.A_5.A_5^3.A_5^3.A_4$

Order 1791590400000000: $D_5^2.A_5^6.C_4^2.D_{12}$, $D_5.A_5.A_5^2.A_5^4.C_4\wr C_2.C_2$

Order 1612431360000000: $(C_3\times A_5).A_5^6.C_4^2.D_6$ x 4, $(S_3\times A_5).A_5^6.C_4^2.S_3$ x 2, $A_4.A_5.A_5^6.C_4^2.C_3$, $A_4.A_5.A_5^3.A_5^3.S_4.C_2$, $(S_3\times A_5).A_5^6.C_4^2.C_6$

Order 1343692800000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2^2$

Order 671846400000000: $A_5^2.A_5^2.A_5^2.A_5^2.C_4$

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 660451885056: $C_2^{16}.C_3^8.C_4^3.S_4$

Order 153600000000: $C_5^8.C_4^3.C_2.C_2^6.S_4.C_2$

Order 2579890176: $C_3^8.C_2^6.C_2^5.C_2^4.D_6$

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

Order 390625: $C_5^8$

Order 19683: $C_3^8.C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 257989017600000000: $A_5^8.C_4^3.S_4$ ($C_1$)

Order 128994508800000000: $A_5^8.C_4^3.A_4$ ($C_2$)

Order 42998169600000000: $A_5^8.C_4^3.C_2^2$ ($S_3$)

Order 10749542400000000: $A_5^8.C_2.C_2^5$ ($S_4$) x 2, $A_5^2.A_5^2.A_5^2.A_5^2.C_4^3$ ($S_4$)

Order 2687385600000000: $A_5^2.A_5^2.A_5^2.A_5^2.C_2^3.C_2$ ($C_2^2:S_4$)

Order 1343692800000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2^2$ ($C_2^3:S_4$)

Order 671846400000000: $A_5^2.A_5^2.A_5^2.A_5^2.C_4$ ($C_4^2:S_4$)

Order 335923200000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5$ ($C_2^5:S_4$)

Order 167961600000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.A_5$ ($C_4^3:S_4$)

Order 1: $C_1$ ($A_5^8.C_4^3.S_4$)

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 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 $6033 \times 6033$ character table is not available for this group.

Rational character table

The $5270 \times 5270$ rational character table is not available for this group.