Abstract group 2432902008176640000.b: $C_2.A_{20}$

• Label: 243...000.b
• Order: \( 2^{18} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• Exponent: \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{18} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $22$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$C_2.A_{20}$
Order:	\(2432902008176640000\)\(\medspace = 2^{18} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Exponent:	\(232792560\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Automorphism group:	Group of order \(2432902008176640000\)\(\medspace = 2^{18} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Composition factors:	$C_2$, $A_{20}$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	24	26	28	30	33	34	35	36	38	39	40	42	44	45	51	52	55	56	60	63	65	66	70	72	77	78	84	90	91	99	102	105	110	120	126	130	132	140	154	165	168	180	182	198	210	330
Elements	1	23596729471	3044269834280	294446829974400	189239120970624	8603851448997080	34479959558400	13959955250841600	8470676211379200	34141660851659904	609493248000	131396469228460800	37132204032000	17649640228550400	61842055316670720	152056375511040000	23851980472320000	171569240896435200	128047474114560000	44250550689945600	3030432354892800	1613328627456000	108759194159616000	7834895050752000	53168011573248000	190088668518577920	3515557054464000	23851980472320000	6125354900121600	45616912653312000	128047474114560000	12996271411200000	44603203483238400	142770777803596800	23038844774400000	11488703927500800	47703960944640000	46786577080320000	1843107581952000	28963119144960000	80496017060659200	14481559572480000	18714630832128000	49593246603264000	41967214842009600	33790305669120000	15798064988160000	28591797104640000	25342729251840000	18246765061324800	26735186903040000	24574767759360000	47703960944640000	4441011602227200	12901753073664000	40548366802944000	24135932620800000	18714630832128000	18431075819520000	13033403615232000	15798064988160000	14744860655616000	14481559572480000	13516122267648000	26735186903040000	24574767759360000	21818883089203200	14744860655616000	2432902008176640000
Conjugacy classes	1	11	6	24	4	66	2	20	5	26	1	82	1	14	13	4	1	21	2	26	6	5	30	3	12	61	3	1	3	12	2	2	10	26	4	4	2	2	1	4	30	3	1	7	9	2	1	4	12	6	2	2	2	3	3	6	5	1	2	4	1	2	2	2	2	2	7	2	648
Divisions	1	11	6	24	4	66	2	20	5	26	1	82	1	14	12	4	1	21	1	26	6	5	30	3	12	60	3	1	3	12	1	2	10	26	4	4	1	2	1	4	30	2	1	7	9	2	1	4	12	6	1	1	1	3	3	6	4	1	2	4	1	1	2	2	1	1	7	1	634
Autjugacy classes	1	11	6	24	4	66	2	20	5	26	1	82	1	14	12	4	1	21	1	26	6	5	30	3	12	60	3	1	3	12	1	2	10	26	4	4	1	2	1	4	30	2	1	7	9	2	1	4	12	6	1	1	1	3	3	6	4	1	2	4	1	1	2	2	1	1	7	1	634

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$40$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	19	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $22$ $\langle(1,2,3,4,6,10,7)(5,8,12,17,15,18,19,20,13,16,9,14,11), (2,4,7,6,11,8,13,17,12,18,10,16,3,5,9,15,19)(21,22)\rangle$
Transitive group:	40T315648				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:	$A_{20}$ . $C_2$	$C_2$ . $A_{20}$			more information

Elements of the group are displayed as permutations of degree 22.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 4 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $A_{20}$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_2^5.C_2^5.C_2^4.C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4.C_3^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^4$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}$

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 2150400 (order at least 1131371841600) are shown, as well as all normal subgroups of any index.

Order 2432902008176640000: $C_2.A_{20}$

Order 1216451004088320000: $A_{20}$

Order 121645100408832000: $C_2.A_{19}$

Order 60822550204416000: $A_{19}$

Order 12804747411456000: $C_2.S_{18}$

Order 6402373705728000: $S_{18}$ x 2, $C_2.A_{18}$

Order 3201186852864000: $A_{18}$

Order 2134124568576000: $C_6.A_{17}.C_2$

Order 1067062284288000: $C_3.A_{17}.C_2$ x 2, $C_6.A_{17}$

Order 711374856192000: $C_2\times S_{17}$

Order 533531142144000: $C_3\times A_{17}$

Order 502146957312000: $(C_2\times A_4).A_{16}.C_2$

Order 355687428096000: $S_{17}$ x 2, $C_2\times A_{17}$

Order 251073478656000: $A_4.A_{16}.C_2$ x 2, $(C_2\times A_4).A_{16}$

Order 177843714048000: $A_{17}$

Order 167382319104000: $C_2^3.A_{16}.C_2$

Order 156920924160000: $C_2.A_{15}.S_5$

Order 125536739328000: $C_6.A_{16}.C_2$, $A_4.A_{16}$

Order 83691159552000: $C_2^2.A_{16}.C_2$ x 6, $C_2^3.A_{16}$

Order 78460462080000: $A_{15}.S_5$ x 2, $C_2.A_{15}.A_5$

Order 62768369664000: $C_3.A_{16}.C_2$ x 2, $C_6.A_{16}$, $C_2.A_{14}.A_6.C_2$

Order 41845579776000: $C_2.S_{16}$ x 4, $C_2^2.A_{16}$ x 3, $C_2.A_{16}.C_2$ x 2

Order 39230231040000: $A_{15}\times A_5$

Order 31384184832000: $A_{14}.A_6.C_2$ x 2, $C_3\times A_{16}$, $C_2.A_{14}.A_6$, $C_2.A_{13}.A_7.C_2$, $(C_2\times A_4).A_{15}.C_2$

Order 26336378880000: $A_{10}\wr C_2.C_2^2$

Order 26153487360000: $D_{10}.A_{15}.C_2$

Order 20922789888000: $C_2.A_{16}$ x 3, $S_{16}$ x 2

Order 19313344512000: $C_2.A_{12}.A_8.C_2$

Order 15692092416000: $A_{13}.A_7.C_2$ x 2, $A_4.A_{15}.C_2$ x 2, $D_6.A_{15}.C_2$, $C_2.A_{13}.A_7$, $A_{14}\times A_6$, $(C_2\times A_4).A_{15}$

Order 14485008384000: $C_2.A_{11}.A_9.C_2$

Order 13168189440000: $A_{10}^2.C_2^2$ x 4, $C_2\times A_{10}\wr C_2$ x 2, $C_2.A_{10}.A_{10}.C_2$

Order 13076743680000: $D_5.A_{15}.C_2$ x 2, $D_{10}.A_{15}$

Order 10461394944000: $C_2^3.A_{15}.C_2$, $C_2.A_{14}.S_5$, $(C_2\times A_5).A_{14}.C_2$, $A_{16}$

Order 9656672256000: $A_{12}.A_8.C_2$ x 2, $C_2.A_{12}.A_8$

Order 7846046208000: $S_3\times S_{15}$ x 4, $C_6.A_{15}.C_2$ x 2, $D_6.A_{15}$, $A_{13}\times A_7$, $A_4.A_{15}$

Order 7242504192000: $A_{11}.A_9.C_2$ x 2, $C_2.A_{11}.A_9$

Order 6584094720000: $A_{10}\wr C_2$ x 4, $A_{10}.A_{10}.C_2$ x 2, $C_2.A_{10}.A_{10}$

Order 6538371840000: $A_{15}\times D_5$ x 2, $C_{10}.A_{15}$

Order 6276836966400: $(C_2\times C_3:S_3.C_2).A_{14}.C_2$

Order 5230697472000: $C_2^2.A_{15}.C_2$ x 6, $A_{14}.S_5$ x 2, $A_5.A_{14}.C_2$ x 2, $C_2^3.A_{15}$, $C_2.A_{14}.A_5$, $(C_2\times A_5).A_{14}$

Order 4828336128000: $A_{12}\times A_8$

Order 4483454976000: $C_2.A_6.A_{13}.C_2$

Order 4184557977600: $(C_2\times S_4).A_{14}.C_2$ x 2

Order 3923023104000: $C_3.S_{15}$ x 2, $C_3.S_{15}$ x 2, $S_3\times A_{15}$ x 2, $C_6.A_{15}$

Order 3621252096000: $A_{11}\times A_9$

Order 3292047360000: $A_{10}^2$

Order 3269185920000: $C_5\times A_{15}$

Order 3138418483200: $C_3:S_3.C_2.A_{14}.C_2$ x 4, $(C_2\times C_3:S_3).A_{14}.C_2$ x 2, $(C_2\times C_3:S_3.C_2).A_{14}$

Order 2615348736000: $C_2\times S_{15}$ x 4, $C_2^2.A_{15}$ x 3, $C_2.A_{15}.C_2$ x 2, $A_{14}\times A_5$, $A_5\times A_{14}$

Order 2414168064000: $C_2.A_7.A_{12}.C_2$

Order 2241727488000: $A_6.A_{13}.C_2$ x 2, $C_2.A_6.A_{13}$

Order 2092278988800: $S_4.A_{14}.C_2$ x 8, $(C_2\times A_4).A_{14}.C_2$ x 4, $(C_2\times S_4).A_{14}$ x 2

Order 1961511552000: $C_3.A_{15}$

Order 1743565824000: $D_{10}.A_{14}.C_2$

Order 1609445376000: $C_2.A_8.A_{11}.C_2$

Order 1569209241600: $C_3:S_3.A_{14}.C_2$ x 6, $C_3:S_3.C_2\times A_{14}$ x 2, $(C_3\times C_6).A_{14}.C_2$ x 2, $(C_2\times C_3:S_3).A_{14}$

Order 1494484992000: $C_2.A_{13}.S_5.C_2$

Order 1394852659200: $(C_2\times D_4).A_{14}.C_2$

Order 1316818944000: $C_2.A_9.A_{10}.C_2$

Order 1307674368000: $C_2.A_{15}$ x 3, $S_{15}$ x 2

Order 1207084032000: $A_7.A_{12}.C_2$ x 2, $C_2.A_7.A_{12}$

Order 4976640000: $C_2\times A_5^4.Q_8.S_4$

Order 3715891200: $C_2^{10}.S_{10}$

Order 955514880: $C_2^{10}.(C_3^5.C_2\wr S_5)$

Order 262144: $C_2\times C_2^5.C_2^5.C_2^4.C_2^3$

Order 6840: $C_2\times \PSL(2,19)$

Order 6561: $C_3^4.C_3^4$

Order 625: $C_5^4$

Order 49: $C_7^2$

Order 19: $C_{19}$

Order 17: $C_{17}$

Order 13: $C_{13}$

Order 11: $C_{11}$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 2432902008176640000: $C_2.A_{20}$ ($C_1$)

Order 1216451004088320000: $A_{20}$ ($C_2$)

Order 2: $C_2$ ($A_{20}$)

Order 1: $C_1$ ($C_2.A_{20}$)

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 4 larger groups in the database.

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

Character theory

Complex character table

The $648 \times 648$ character table is not available for this group.

Rational character table

The $634 \times 634$ rational character table is not available for this group.