Abstract group 12383472844800000000.a: $A_5^8.C_2^6.A_4^2.C_4.C_2$

• Label: 123...000.a
• Order: \( 2^{29} \cdot 3^{10} \cdot 5^{8} \)
• Exponent: \( 2^{4} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{30} \cdot 3^{10} \cdot 5^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: not computed
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$A_5^8.C_2^6.A_4^2.C_4.C_2$
Order:	\(12383472844800000000\)\(\medspace = 2^{29} \cdot 3^{10} \cdot 5^{8} \)
Exponent:	\(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \)
Automorphism group:	Group of order \(24766945689600000000\)\(\medspace = 2^{30} \cdot 3^{10} \cdot 5^{8} \)
Composition factors:	$C_2$ x 13, $C_3$ x 2, $A_5$ x 8
Derived length:	$4$

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:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Permutation group:	Degree $40$ $\langle(1,34,4,35,3,33,5,31,2,32)(6,19,40,26,11,22,10,18,37,28,15,21)(7,20,36,30,12,25,9,16,38,29,14,24,8,17,39,27,13,23) \!\cdots\! \rangle$
Transitive group:	40T315722				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(A_5^8.C_2^6.A_4^2.C_4)$ . $C_2$	$(A_5^8.C_2^6.C_2^4.D_6)$ . $C_6$	$(A_5^4.A_5^4.C_2^6.A_4^2)$ . $D_4$	$(A_5^8.C_2^6.A_4^2.C_2^2)$ . $C_2$	all 23

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

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 25 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:	$P_{ 2 } \simeq$ $C_2^{12}.C_2^6.C_2^6.C_2^5$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^8.C_3^2$
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 90 (order at least 137594142720000000) are shown, as well as all normal subgroups of any index.

Order 12383472844800000000: $A_5^8.C_2^6.A_4^2.C_4.C_2$

Order 6191736422400000000: $A_5^4.A_5^4.C_2^4.C_2^6.C_6^2$, $A_5^8.C_2^6.A_4^2.C_4$, $A_5^8.C_2^6.A_4^2.C_2^2$

Order 4127824281600000000: $A_5^8.C_2^6.C_2^4.C_3.D_4$, $A_5^8.C_2^6.C_2^4.C_{12}.C_2$

Order 3095868211200000000: $A_5^4.A_5^4.C_2^6.A_4^2.C_2$ x 2, $A_5^8.C_2^6.A_4\wr C_2$

Order 2063912140800000000: $A_5^4.A_5^4.C_2^6.C_2^4.C_6.C_2$ x 2, $A_5^4.A_5^4.C_2^6.A_4.C_2^4$, $A_5^8.C_2^6.C_2^4.C_6.C_2$, $A_5^8.C_2^6.C_2^4.D_6$, $A_5^8.C_2^6.C_2^4.C_{12}$, $A_5^8.C_2^4.C_2^5.A_4.C_2$

Order 1547934105600000000: $A_5^4.A_5^4.C_2^6.A_4^2$, $A_5.A_5^3.A_5^4.C_2.C_2^6.C_6^2.C_2$

Order 1375941427200000000: $A_5^8.C_2^6.C_2^6.C_2$

Order 1031956070400000000: $A_5^4.A_5^4.C_2^6.A_4.C_2^3$ x 4, $A_5^4.A_5^4.C_2^4.C_2^6.C_6$ x 4, $A_5^4.A_5^4.C_2^5.A_4.C_4.C_2^2$, $A_5^4.A_5^4.C_2^4.A_4.C_2^5$, $A_5^2.A_5^2.A_5^4.C_2.C_2^6.C_6.C_2^3$, $A_5^8.C_2^6.C_2^4.S_3$, $A_5^8.C_2^4.C_2^5.A_4$, $A_5^8.C_2^6.A_4.D_4$, $A_5^8.C_2^6.(D_4\times A_4)$

Order 773967052800000000: $A_5.A_5^3.A_5^4.C_2.C_2^6.C_6^2$ x 7, $A_5^4.A_5^4.C_2^2.C_2^6.C_3.C_6$, $A_5^4.A_5^4.C_2.C_2^6.C_6^2$, $A_5^2.A_5^6.C_2^4.C_6^2.C_2^3$

Order 687970713600000000: $A_5^8.C_2^6.C_2^5.C_2$ x 2, $A_5^8.C_2^6.C_2^4.C_2^2$, $A_5^4.A_5^4.C_2^6.C_2^6$, $A_5^8.C_2^6.C_2^5.C_2$, $A_5^8.C_2^6.C_2^5.C_2$, $A_5^8.C_2^6.C_2^5.C_2$

Order 515978035200000000: $A_5^4.A_5^4.C_2^4.A_4.C_2^4$ x 5, $A_5^2.A_5^2.A_5^4.C_2^5.A_4.C_2^3$ x 4, $A_5^4.A_5^4.C_2^5.A_4.C_4.C_2$ x 3, $A_5^4.A_5^4.C_2^6.A_4.C_2^2$ x 2, $A_5^4.A_5^4.C_2^4.C_2^6.C_3$ x 2, $A_5^2.A_5^2.A_5^4.C_2^4.A_4.C_2^4$ x 2, $A_5^2.A_5^2.A_5^4.C_2.C_2^6.C_6.C_2^2$ x 2, $A_5.A_5^3.A_5^4.C_2.C_2^6.C_6.C_2^2$ x 2, $A_5^4.A_5^4.C_2^5.A_4.C_2^2.C_2$, $A_5^4.A_5^4.C_2.C_2^5.C_6.C_2^3$, $A_5^2.A_5^2.A_5^4.C_2^4.A_4.C_2^3.C_2$, $A_5^2.A_5^2.A_5^4.C_2.C_2^5.C_6.C_2^3$, $A_5.A_5^3.A_5^4.C_2^5.C_6.C_2^4$, $A_5.A_5.A_5.A_5.A_5^4.C_2\wr A_4.C_2^4$, $A_5^8.C_2^6.A_4.C_4$, $A_5^8.C_2^6.S_4.C_2$, $A_5^8.C_2^6.A_4.C_4$, $A_5^8.C_2^5.C_2^3.A_4$

Order 386983526400000000: $A_5^2.A_5^6.C_2^4.C_6^2.C_2^2$ x 6, $A_5.A_5^3.A_5^4.C_2^5.A_4.C_6$ x 4, $A_5^4.A_5^4.C_2^5.A_4.C_6$ x 2, $A_5.A_5^3.A_5^4.C_2^3.A_4^2.C_2$ x 2, $A_5^4.A_5^4.C_2^5.(C_6\times A_4)$, $A_5^4.A_5^4.C_2^2.C_2^6.C_3^2$, $A_5.A_5^3.A_5^4.C_2^5.(C_6\times A_4)$, $A_5.A_5^3.A_5^4.C_2.C_2^4.C_6^2.C_2$, $A_5.A_5.A_5^3.A_5^3.A_4^2.C_2^4$

Order 343985356800000000: $A_5^4.A_5^4.C_2^6.C_2^5$ x 8, $A_5^8.C_2^4.C_4^2.D_4$ x 2, $A_5^8.C_2^4.C_4^2.D_4$ x 2, $A_5^8.C_2^6.C_2^4.C_2$, $A_5^8.C_2^6.C_2^3.C_2^2$, $A_5^8.C_2^6.C_2^2:D_4$, $A_5^8.C_2^6.C_2.C_2^4$, $A_5^8.C_2^5.C_2.C_2^5$, $A_5^8.C_2^4.C_2^6.C_2$, $A_5^4.A_5^4.C_2^6.C_2^4.C_2$, $A_5^2.A_5^2.A_5^4.C_2^6.C_2^5$, $A_5^8.C_2^5.C_2^3.D_4$, $A_5^8.C_2^6.C_2^4.C_2$, $A_5^8.C_2^6.C_2^4.C_2$, $A_5^8.C_2^6.C_4:D_4$, $A_5^8.C_2^6.C_2^3.C_2^2$, $A_5^8.C_2^6.(C_4\times D_4)$, $A_5^8.C_2^5.C_2^3.D_4$, $A_5^8.C_2^4.C_4^2.C_2^3$, $A_5^8.C_2^5.C_2^4.C_2^2$

Order 309586821120000000: $S_4.A_5^3.A_5^4.C_2.C_2^6.C_6^2$

Order 257989017600000000: $A_5^4.A_5^4.C_2^4.A_4.C_2^3$ x 17, $A_5^2.A_5^2.A_5^4.C_2^4.A_4.C_2^3$ x 12, $A_5.A_5^3.A_5^4.C_2^5.C_6.C_2^3$ x 12, $A_5.A_5^3.A_5^4.C_2.C_2^6.C_6.C_2$ x 12, $A_5^2.A_5^2.A_5^4.C_2^4.A_4.C_2^2.C_2$ x 7, $A_5^4.A_5^4.C_2^4.C_2^4.C_6$ x 6, $A_5.A_5.A_5.A_5.A_5^4.C_2\wr A_4.C_2^3$ x 6, $A_5^4.A_5^4.C_2^6.A_4.C_2$ x 4, $A_5^2.A_5^2.A_5^4.C_2.C_2^5.C_6.C_2^2$ x 3, $A_5.A_5.A_5.A_5.A_5^4.C_2^3.A_4.C_2^4$ x 3, $A_5^4.A_5^4.C_2.C_2^6.C_6.C_2$ x 2, $A_5^2.A_5^2.A_5^4.C_2^3.A_4.C_2^3.C_2$ x 2, $A_5.A_5^3.A_5^4.C_2^5.A_4.C_2^2$ x 2, $F_5.A_5^3.A_5^4.C_2.C_2^6.C_6^2$, $A_5^4.A_5^4.C_2^5.C_6.C_2^3$, $A_5^4.A_5^4.C_2^3.A_4.C_2^4$, $A_5^4.A_5^4.C_2^3.A_4.C_2.C_2^3$, $A_5^4.A_5^4.C_2^3.(D_4\times A_4).C_2$, $A_5^4.A_5^4.C_2.C_2^5.C_6.C_2^2$, $A_5^2.A_5^6.C_2^6.C_6.C_2^2$, $A_5^2.A_5^6.C_2^4.C_2^4:S_3$, $A_5^2.A_5^2.A_5^4.C_2^3.(D_4\times A_4).C_2$, $A_5^2.A_5^2.A_5^4.C_2\wr A_4.C_2^3$, $A_5.A_5^3.A_5^4.C_2^4.C_6.C_2^4$, $A_5.A_5^3.A_5^4.C_2.A_4.C_2^6$, $A_5.A_5^2.A_5^2.A_5^3.C_2^4.C_6.C_2^4$, $A_5^8.C_2^6.S_4$, $A_5^8.C_2^4.C_2^4.S_3$, $A_5^8.C_2^4.Q_8:A_4$, $A_5^8.Q_8^2.C_2.A_4$, $A_5^8.C_2^6.S_4$, $A_5^8.C_2^6.A_4.C_2$

Order 193491763200000000: $A_5.A_5.A_5^3.A_5^3.A_4^2.C_2^3$ x 9, $A_5.A_5^3.A_5^4.C_2.C_2^4.C_6^2$ x 7, $A_5^2.A_5^6.C_2^4.C_6^2.C_2$ x 6, $A_5^2.A_5^6.A_4^2.C_2^2.C_2$ x 3, $A_5^4.A_5^4.C_2^3.A_4^2$, $A_5^2.A_5^6.A_4^2.C_2^3$, $A_5.A_5^3.A_5^4.C_2^3.A_4^2$, $A_5.A_5^3.A_5^4.C_2.A_4^2.C_2^2$, $A_5.A_5^3.A_5^4.A_4^2.C_2^3$, $A_5^8.A_4^2.C_4.C_2$, $A_5^8.Q_8^2.C_3.C_6$

Order 171992678400000000: $A_5^4.A_5^4.C_2^6.C_2^4$ x 38, $A_5^2.A_5^2.A_5^4.C_2^5.C_2^5$ x 14, $A_5^4.A_5^4.C_2^5.C_2^4.C_2$ x 8, $A_5^4.A_5^4.C_2^5.C_2^5$ x 7, $A_5^2.A_5^2.A_5^4.C_2^6.C_2^4$ x 7, $A_5^8.C_2^5.C_2^4.C_2$ x 2, $A_5^8.C_2^4.C_4^2.C_2^2$ x 2, $A_5^8.C_2^4.C_4^2.C_4$ x 2, $A_5^8.C_2^5.C_2^4.C_2$ x 2, $A_5^8.C_2^5.C_2^3.C_4$ x 2, $A_5^8.C_2^6.C_4.C_2^2$, $A_5^8.C_2^6.C_2^4$, $A_5^8.C_2^6.C_2.C_2^3$, $A_5^8.C_2^5.C_2^4.C_2$, $A_5^8.C_2^5.C_2.C_2^4$, $A_5^8.C_2^4.C_2^5.C_2$, $A_5^4.A_5^4.C_2^6.C_2^3.C_2$, $A_5^4.A_5^4.C_2^4.C_2^6$, $A_5^2.A_5^2.A_5^2.A_5^2.C_2^6.C_2^4$, $A_5.A_5.A_5.A_5.A_5^4.C_2^5.C_2^5$, $A_5^8.C_2^4.D_4^2$, $A_5^8.C_2^4.C_2^3.D_4$, $A_5^8.C_2^5.C_2.C_4^2$, $A_5^8.C_2^6.C_2.C_2^3$, $A_5^8.C_2^4.C_2^5.C_2$, $A_5^8.C_2^4.C_4^2.C_2^2$, $A_5^8.C_2^6.C_4^2$, $A_5^8.C_2^4.C_4^2.C_2^2$, $A_5^8.C_2^4.C_4^2.C_2^2$, $A_5^8.C_4^2.D_4.C_2^3$, $A_5^8.C_2^3.C_2^6.C_2$, $A_5^8.C_2^4.C_4^2.C_2^2$, $A_5^8.C_4.C_2^4.C_2^4$, $A_5^8.C_2^5.C_2^3.C_4$, $A_5^8.C_2^4.C_2^5.C_2$, $A_5^8.C_2^5.C_2^4.C_2$, $A_5^8.C_2^5.C_2^4.C_2$, $A_5^8.C_2^4.C_2^3.D_4$, $A_5^8.C_2.C_2^5.C_2^4$, $A_5^8.C_2^4.C_4^2.C_2^2$, $A_5^8.C_2^4.C_4^2.C_2^2$, $A_5^8.C_2^4.D_4^2$

Order 154793410560000000: $A_4.A_5^3.A_5^4.C_2.C_2^6.C_6^2$ x 4, $S_4.A_5^3.A_5^4.C_2^5.A_4.C_6$ x 3, $D_6.A_5^3.A_5^4.C_2.C_2^6.C_6^2$

Order 42998169600000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2^6.C_2$

Order 21499084800000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2^6$

Order 10749542400000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2^5$

Order 671846400000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.A_5.C_2^2$

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 31701690482688: $C_2^{16}.C_3^8.C_2^6.A_4^2.C_4.C_2$

Order 7372800000000: $C_5^8.C_2^4.C_2^6.C_2^6.D_{12}.C_{12}$

Order 123834728448: $C_3^8.C_2^4.C_2^4.C_2^4.C_2^4.C_6^2.C_2^3$

Order 536870912: $C_2^{12}.C_2^6.C_2^6.C_2^5$

Order 390625: $C_5^8$

Order 59049: $C_3^8.C_3^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 12383472844800000000: $A_5^8.C_2^6.A_4^2.C_4.C_2$ ($C_1$)

Order 6191736422400000000: $A_5^4.A_5^4.C_2^4.C_2^6.C_6^2$ ($C_2$), $A_5^8.C_2^6.A_4^2.C_4$ ($C_2$), $A_5^8.C_2^6.A_4^2.C_2^2$ ($C_2$)

Order 4127824281600000000: $A_5^8.C_2^6.C_2^4.C_3.D_4$ ($C_3$)

Order 3095868211200000000: $A_5^4.A_5^4.C_2^6.A_4^2.C_2$ ($C_2^2$)

Order 2063912140800000000: $A_5^4.A_5^4.C_2^6.C_2^4.C_6.C_2$ ($C_6$), $A_5^4.A_5^4.C_2^6.C_2^4.C_6.C_2$ ($S_3$), $A_5^8.C_2^6.C_2^4.C_6.C_2$ ($C_6$), $A_5^8.C_2^6.C_2^4.D_6$ ($C_6$)

Order 1547934105600000000: $A_5^4.A_5^4.C_2^6.A_4^2$ ($D_4$)

Order 1031956070400000000: $A_5^4.A_5^4.C_2^4.C_2^6.C_6$ ($C_2\times C_6$), $A_5^4.A_5^4.C_2^4.C_2^6.C_6$ ($D_6$)

Order 687970713600000000: $A_5^4.A_5^4.C_2^6.C_2^6$ ($C_3\times S_3$)

Order 515978035200000000: $A_5^4.A_5^4.C_2^4.C_2^6.C_3$ ($C_3:D_4$), $A_5^4.A_5^4.C_2^4.C_2^6.C_3$ ($C_3\times D_4$)

Order 343985356800000000: $A_5^4.A_5^4.C_2^6.C_2^5$ ($C_6\times S_3$)

Order 171992678400000000: $A_5^4.A_5^4.C_2^4.C_2^6$ ($C_6\wr C_2$)

Order 42998169600000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2^6.C_2$ ($A_4\wr C_2$)

Order 21499084800000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2^6$ ($A_4^2:C_2^2$)

Order 10749542400000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5.C_2^5$ ($A_4^2:D_4$)

Order 671846400000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.A_5.C_2^2$ ($C_2^6.A_4\wr C_2$)

Order 335923200000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.S_5$ ($C_2^5.C_2^4.C_6^2.C_2$)

Order 167961600000000: $A_5.A_5.A_5.A_5.A_5.A_5.A_5.A_5$ ($C_2^8.A_4\wr C_2$)

Order 1: $C_1$ ($A_5^8.C_2^6.A_4^2.C_4.C_2$)

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 1 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.