Abstract group 520203639383182540800000000.a: $A_{10}\wr C_2^2.D_6$

• Label: 520...000.a
• Order: \( 2^{32} \cdot 3^{17} \cdot 5^{8} \cdot 7^{4} \)
• Exponent: \( 2^{5} \cdot 3^{3} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{32} \cdot 3^{17} \cdot 5^{8} \cdot 7^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $40$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$A_{10}\wr C_2^2.D_6$
Order:	\(520\!\cdots\!000\)\(\medspace = 2^{32} \cdot 3^{17} \cdot 5^{8} \cdot 7^{4} \)
Exponent:	\(30240\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 7 \)
Automorphism group:	Group of order \(520\!\cdots\!000\)\(\medspace = 2^{32} \cdot 3^{17} \cdot 5^{8} \cdot 7^{4} \)
Composition factors:	$C_2$ x 4, $C_3$, $A_{10}$ x 4
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:	$40$
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 $40$ $\langle(1,12,25,8,20,27,6,13,30,3,15,28,2,17,24,7,14,22,4,11,21)(5,19,29,9,18,26) \!\cdots\! \rangle$
Transitive group:	40T315823				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:	$(A_{10}^4.S_4)$ . $C_2$	$(A_{10}^4.S_4)$ . $C_2$	$(A_{10}^4.C_2^3)$ . $S_3$	$(A_{10}\wr A_4)$ . $C_2^2$	all 8

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

Homology

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

Subgroups

There are 10 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 $A_{10}\wr A_4$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^6.C_2^6.C_2^5$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^9.C_3^5.C_3^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^8$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^4$

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 6272 (order at least 82940631279206400000000) are shown, as well as all normal subgroups of any index.

Order 520203639383182540800000000: $A_{10}\wr C_2^2.D_6$

Order 260101819691591270400000000: $A_{10}^4.S_4$, $A_{10}^4.S_4$, $A_{10}\wr A_4.C_2$

Order 173401213127727513600000000: $A_{10}\wr C_4.C_2^2$

Order 130050909845795635200000000: $A_{10}.A_{10}^3.D_6$, $A_{10}\wr A_4$

Order 86700606563863756800000000: $A_{10}^4.D_4$ x 2, $A_{10}\wr D_4$ x 2, $A_{10}^2.A_{10}^2.C_2^3$, $A_{10}^4.C_2^3$, $A_{10}^4.C_2^2.C_2$

Order 65025454922897817600000000: $A_{10}.A_{10}^3.S_3$ x 2, $A_{10}.A_{10}^3.C_6$

Order 43350303281931878400000000: $A_{10}^2.A_{10}^2.C_2^2$ x 4, $A_{10}\wr C_2^2$ x 2, $A_{10}\wr C_4$ x 2, $A_{10}.A_{10}.A_{10}^2.C_2^2$

Order 32512727461448908800000000: $A_{10}.A_{10}^3.C_3$

Order 21675151640965939200000000: $A_{10}^2.A_{10}^2.C_2$ x 2, $A_{10}.A_{10}.A_{10}^2.C_2$ x 2, $A_{10}.A_{10}.A_{10}.A_{10}.C_2$

Order 13005090984579563520000000: $A_9.A_{10}^3.D_6$

Order 10837575820482969600000000: $A_{10}.A_{10}.A_{10}.A_{10}$

Order 6502545492289781760000000: $A_{10}^3.S_3\times A_9$, $A_9.A_{10}^3.S_3$, $A_9.A_{10}^3.C_6$

Order 4335030328193187840000000: $A_9.A_{10}.A_{10}^2.C_2^2$

Order 3251272746144890880000000: $A_{10}^3.C_3\times A_9$

Order 2890020218795458560000000: $A_8.A_{10}^3.D_6.C_2$

Order 2167515164096593920000000: $A_9.A_{10}.A_{10}^2.C_2$ x 2, $A_9.A_{10}.A_{10}.A_{10}.C_2$

Order 1445010109397729280000000: $A_8.A_{10}^3.D_6$ x 5, $A_{10}^3.S_3\times S_8$, $A_8.A_{10}^3.C_6.C_2$

Order 1083757582048296960000000: $C_3.A_7.A_{10}^3.D_6.C_2$, $A_9.A_{10}.A_{10}.A_{10}$

Order 1032150078141235200000000: $A_5^2.A_{10}^3.D_6.C_2^2$

Order 963340072931819520000000: $A_{10}.A_8.A_{10}^2.C_2^3$

Order 867006065638637568000000: $A_9^2.A_{10}^2.C_2^3$

Order 722505054698864640000000: $A_8.A_{10}^3.S_3$ x 3, $A_8.A_{10}^3.C_6$ x 2, $A_{10}^3.S_3\times A_8$, $A_{10}^3.C_3\times S_8$

Order 619290046884741120000000: $A_4.A_6.A_{10}^3.D_6.C_2$

Order 541878791024148480000000: $C_3.A_7.A_{10}^3.D_6$ x 5, $C_3.A_7.A_{10}^3.C_6.C_2$, $A_{10}^3.S_3\times C_3.A_7.C_2$

Order 516075039070617600000000: $A_5^2.A_{10}^3.D_6.C_2$ x 2, $A_{10}^3.S_3\times A_5^2.C_4$, $A_5^2.A_{10}^3.D_{12}$, $A_5^2.A_{10}^3.C_6.C_2^2$, $A_5^2.A_{10}^3.C_3.D_4$, $A_5.A_5.A_{10}^3.D_6.C_2$

Order 481670036465909760000000: $A_{10}.A_8.A_{10}^2.C_2^2$ x 6, $A_{10}.A_{10}.A_{10}.A_8.C_2^2$

Order 433503032819318784000000: $A_9^2.A_{10}^2.C_2^2$ x 5, $A_{10}.A_{10}.A_9^2.C_2^2$, $A_9.A_9.A_{10}^2.C_2^2$

Order 361252527349432320000000: $C_3.A_7.A_{10}.A_{10}^2.C_2^3$, $A_{10}^3.C_3\times A_8$, $A_7.A_{10}^3.D_6.C_2$

Order 344050026047078400000000: $A_{10}.A_5^2.A_{10}^2.C_2.C_2^3$

Order 309645023442370560000000: $A_4.A_6.A_{10}^3.D_6$ x 5, $A_{10}^3.S_3\times A_4.A_6.C_2$, $A_4.A_6.A_{10}^3.C_6.C_2$

Order 270939395512074240000000: $C_3.A_7.A_{10}^3.S_3$ x 4, $C_3.A_7.A_{10}^3.C_6$ x 2, $A_{10}^3.C_3\times C_3.A_7.C_2$

Order 258037519535308800000000: $A_5.A_5.A_{10}^3.D_6$ x 4, $A_5^2.A_{10}^3.D_6$ x 3, $A_5^2.A_{10}^3.C_6.C_2$ x 2, $A_{10}^3.C_3\times A_5^2.C_4$, $A_5.A_5.A_{10}^3.C_6.C_2$

Order 240835018232954880000000: $A_{10}.A_8.A_{10}^2.C_2$ x 4, $A_{10}.A_{10}.A_{10}.A_8.C_2$ x 3

Order 216751516409659392000000: $A_{10}.A_{10}.A_9^2.C_2$ x 2, $A_9^2.A_{10}^2.C_2$ x 2, $A_9.A_9.A_{10}^2.C_2$ x 2, $A_9.A_9.A_{10}.A_{10}.C_2$

Order 206430015628247040000000: $C_2^2.A_6.A_{10}^3.D_6.C_2$, $A_4.A_{10}.A_6.A_{10}^2.C_2^3$

Order 180626263674716160000000: $C_3.A_7.A_{10}.A_{10}^2.C_2^2$ x 6, $A_7.A_{10}^3.D_6$ x 6, $C_3.A_7.A_{10}.A_{10}.A_{10}.C_2^2$, $A_{10}^3.S_3\times S_7$, $A_7.A_{10}^3.C_6.C_2$

Order 172025013023539200000000: $A_{10}.A_5^2.A_{10}^2.D_4$ x 3, $A_{10}.A_{10}.A_{10}.A_5^2.D_4$, $A_{10}.A_5^2.A_{10}^2.C_2^3$, $A_{10}.A_5^2.A_{10}^2.C_2^2.C_2$, $A_5.A_5.A_{10}.A_{10}^2.C_2^3$

Order 154822511721185280000000: $A_4.A_6.A_{10}^3.S_3$ x 4, $A_4.A_6.A_{10}^3.C_6$ x 2, $C_3.A_6.A_{10}^3.D_6.C_2$, $A_{10}^3.C_3\times A_4.A_6.C_2$

Order 137620010418831360000000: $C_2^4.A_5.A_{10}^3.D_6.C_2$

Order 135469697756037120000000: $C_3.A_7.A_{10}^3.C_3$

Order 130050909845795635200000: $A_{10}.A_9^3.D_6$

Order 129018759767654400000000: $A_5.A_5.A_{10}^3.S_3$ x 3, $A_5.A_5.A_{10}^3.C_6$ x 2, $A_5^2.A_{10}^3.S_3$, $A_5^2.A_{10}^3.C_6$

Order 120417509116477440000000: $A_{10}.A_{10}.A_{10}.A_8$, $A_7.A_{10}.A_{10}^2.C_2^3$

Order 108375758204829696000000: $A_9.A_9.A_{10}.A_{10}$

Order 103215007814123520000000: $C_2.A_6.A_{10}^3.D_6.C_2$ x 8, $C_2^2.A_6.A_{10}^3.D_6$ x 7, $A_4.A_{10}.A_6.A_{10}^2.C_2^2$ x 6, $A_4.A_5.A_{10}^3.D_6.C_2$ x 2, $C_2^2.A_6.A_{10}^3.C_6.C_2$, $A_4.A_{10}.A_{10}.A_{10}.A_6.C_2^2$

Order 96334007293181952000000: $C_2^3.\PSL(2,7).A_{10}^3.D_6$, $A_8.A_9.A_{10}^2.C_2^3$

Order 90313131837358080000000: $A_7.A_{10}^3.S_3$ x 5, $C_3.A_7.A_{10}.A_{10}^2.C_2$ x 4, $C_3.A_7.A_{10}.A_{10}.A_{10}.C_2$ x 3, $A_7.A_{10}^3.C_6$ x 3, $A_{10}^3.S_3\times A_7$, $A_{10}^3.C_3\times S_7$

Order 86012506511769600000000: $A_5.A_5.A_{10}.A_{10}^2.C_2^2$ x 4, $A_{10}.A_5^2.A_{10}^2.C_2^2$ x 3, $D_5.A_5.A_{10}^3.D_6.C_2$, $A_{10}.A_{10}.A_{10}.A_5^2.C_4$, $A_{10}.A_{10}.A_{10}.A_5^2.C_2^2$, $A_{10}.A_5^2.A_{10}^2.C_4$, $A_5.A_5.A_{10}.A_{10}.A_{10}.C_2^2$

Order 52020363938318254080000: $A_9\wr C_2^2.D_6$

Order 126859598081556480000: $C_2.A_8^4.C_2\wr A_4.C_2$

Order 2508698106593280000: $C_3:(C_3^3:C_2).A_7^4.C_2\wr A_4.C_2$

Order 2063912140800000000: $A_5^8.C_4^3.(D_4\times S_4)$

Order 267483013447680000: $C_2^8.C_3^4.C_2.A_6^4.C_2\wr A_4.C_2$

Order 652298158080000: $C_2^{17}.A_5^4.C_2\wr A_4.C_2$

Order 12899450880000: $A_6^4.C_2\wr C_2^2.D_6$

Order 4294967296: $C_2^9.C_2^6.C_2^6.C_2^6.C_2^5$

Order 129140163: $C_3^9.C_3^5.C_3^3$

Order 87091200: $S_4\times S_{10}$

Order 390625: $C_5^8$

Order 2401: $C_7^4$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 520203639383182540800000000: $A_{10}\wr C_2^2.D_6$ ($C_1$)

Order 260101819691591270400000000: $A_{10}^4.S_4$ ($C_2$), $A_{10}^4.S_4$ ($C_2$), $A_{10}\wr A_4.C_2$ ($C_2$)

Order 130050909845795635200000000: $A_{10}\wr A_4$ ($C_2^2$)

Order 86700606563863756800000000: $A_{10}^4.C_2^3$ ($S_3$)

Order 43350303281931878400000000: $A_{10}\wr C_2^2$ ($D_6$)

Order 21675151640965939200000000: $A_{10}.A_{10}.A_{10}.A_{10}.C_2$ ($S_4$)

Order 10837575820482969600000000: $A_{10}.A_{10}.A_{10}.A_{10}$ ($C_2\times S_4$)

Order 1: $C_1$ ($A_{10}\wr C_2^2.D_6$)

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 2 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.