Abstract group 107145471557284795514880000.a: $C_2^{21}.A_{21}.C_2$

• Label: 107...000.a
• Order: \( 2^{39} \cdot 3^{9} \cdot 5^{4} \cdot 7^{3} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• Exponent: \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{39} \cdot 3^{9} \cdot 5^{4} \cdot 7^{3} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $42$
• Trans deg.: $42$
• Rank: $2$

Group information

Description:	$C_2^{21}.A_{21}.C_2$
Order:	\(107\!\cdots\!000\)\(\medspace = 2^{39} \cdot 3^{9} \cdot 5^{4} \cdot 7^{3} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Exponent:	\(465585120\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Automorphism group:	Group of order \(107\!\cdots\!000\)\(\medspace = 2^{39} \cdot 3^{9} \cdot 5^{4} \cdot 7^{3} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Composition factors:	$C_2$ x 22, $A_{21}$
Derived length:	$1$

This group is nonabelian, nonsolvable, and rational.

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	32	33	34	35	36	38	39	40	42	44	45	48	51	52	55	56	60	63	65	66	68	70	72	76	77	78	80	84	88	90	91	96	99	102	104	105	110	112	120	126	130	132	136	140	144	154	156	160	165	168	176	180	182	195	198	208	210	220	231	240	252	260	264	280	308	312	315	330	336	360	390	420	440	462	504	560	630	660	840
Elements	1	936222729254911	144902514011469200	599870121955233024000	223607273221947210624	27363347883684770073200	6508287607750902374400	179524926813123347742720	31291602508538575257600	284948261049881571214464	1310654280499200	1902302038996660659240960	399245457752064000	641890888805256097996800	241029070141379040460800	1930993327557300348518400	8206607809228308480000	3021394101219060011827200	352452209070015774720000	3837196277672630132736000	2553131720873202765004800	2414706194800469606400	4184611627705716178944000	25870706416875995136000	1651517157919063434854400	4088274500902493996236800	1562538126877069934592000	2056364139932024832000	1239197779193474580480000	3516284342765459865600	5680290417214363376025600	3876974299770173521920000	7333340567989911552000	4057478100932709612257280	3705074333285702978764800	227692733339497660416000	8113306494254343782400	2768480762799576003379200	262611449895305871360000	623959965156897718272000	24478072929161891020800	1146777666566735108505600	6527597761916416598999040	8968649962942365696000	8585374323500384256000	629164803173356929024000	2166544461636273438720000	949318676649661641523200	1942875319749996183552000	1409808836280063098880000	7247393909448376320000	1074423824193058504704000	1537691644757511084441600	4018344914562060386304000	965594448535505338368000	1128712902141817965772800	147177845545720872960000	1116098662055049953280000	135284686309703024640000	1838280149267141099520000	1137562097863800913920000	4003597961852564275200	815639829054093891993600	340144354150110461952000	3759069996607787237376000	554395436598178086912000	678244571556530356224000	1562538126877069934592000	787834349685917614080000	1920330316589181802905600	744065774703366635520000	572544118846421729280000	1688456950288408903680000	669659197233029971968000	10146351473227726848000	1911456045717560229888000	608781088393663610880000	1558817798003553101414400	1030244918820046110720000	68682994588003074048000	946992804167921172480000	515122459410023055360000	664887912360769172275200	994342444376317231104000	57979151275587010560000	773828405691501300940800	584623108695502356480000	412097967528018444288000	541138745238812098560000	575588024288390047334400	347874907653522063360000	343414972940015370240000	42518044268763807744000	476878519241703161856000	425180442687638077440000	297626309881346654208000	480780962116021518336000	821129729940501037056000	243512435357465444352000	405854058929109073920000	212590221343819038720000	191331199209437134848000	297626309881346654208000	162341623571643629568000	127554132806291423232000	107145471557284795514880000
Conjugacy classes	1	131	7	1030	4	1353	3	1366	7	440	1	6228	1	211	14	448	1	461	1	1656	7	71	4256	49	694	1604	24	3	17	3	878	7	2	1128	683	180	4	802	1	90	2	452	3726	2	1	173	12	221	360	2	1	78	218	1354	90	212	1	10	1	7	34	4	54	80	1438	78	23	198	2	344	40	23	58	2	1	478	10	186	7	1	7	2	276	42	1	156	48	8	42	114	8	8	1	31	42	40	7	232	8	7	8	8	7	8	40	35002
Divisions	1	131	7	1030	4	1353	3	1366	7	440	1	6228	1	211	14	448	1	461	1	1656	7	71	4256	49	694	1604	24	3	17	3	878	7	2	1128	683	180	4	802	1	90	2	452	3726	2	1	173	12	221	360	2	1	78	218	1354	90	212	1	10	1	7	34	4	54	80	1438	78	23	198	2	344	40	23	58	2	1	478	10	186	7	1	7	2	276	42	1	156	48	8	42	114	8	8	1	31	42	40	7	232	8	7	8	8	7	8	40	35002
Autjugacy classes	1	101	7	771	4	1037	3	1023	7	340	1	4650	1	164	14	336	1	363	1	1233	7	56	3190	39	515	1247	18	3	14	3	646	6	2	845	534	133	4	601	1	66	2	338	2759	2	1	138	8	176	266	1	1	64	163	995	67	174	1	7	1	7	25	4	45	60	1070	66	19	140	1	252	28	19	40	1	1	357	7	128	7	1	7	1	224	29	1	116	32	4	31	83	4	4	1	27	31	28	7	160	4	7	4	4	7	4	28	26278

Minimal presentations

Permutation degree:	$42$
Transitive degree:	$42$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Permutation group:	Degree $42$ $\langle(1,16,18,9,4,30,5,40,27,24,31)(2,15,17,10,3,29,6,39,28,23,32)(7,11,8,12) \!\cdots\! \rangle$
Transitive group:	42T9475				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_2^{21}$ . $S_{21}$	$C_2^{20}$ . $(C_2.S_{21})$	$(C_2^{21}.A_{21})$ . $C_2$ (3)	$(C_2^{20}.A_{21})$ . $C_2^2$	all 5

Elements of the group are displayed as permutations of degree 42.

Homology

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

Subgroups

There are 9 normal subgroups (7 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 $C_2$
Commutator:	a subgroup isomorphic to $C_2^{20}.A_{21}$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^6.C_2^6.C_2^6.C_2^6$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^7.C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^4$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^3$
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 15925 (order at least 6728130082090097049600) are shown, as well as all normal subgroups of any index.

Order 107145471557284795514880000: $C_2^{21}.A_{21}.C_2$

Order 53572735778642397757440000: $C_2^{20}.A_{21}.C_2$ x 2, $C_2^{21}.A_{21}$

Order 26786367889321198878720000: $C_2^{20}.A_{21}$

Order 5102165312251656929280000: $C_2^{21}.A_{20}.C_2$

Order 2551082656125828464640000: $C_2^{20}.A_{20}.C_2$ x 6, $C_2^{21}.A_{20}$

Order 1275541328062914232320000: $C_2^{19}.A_{20}.C_2$ x 4, $C_2^{20}.A_{20}$ x 3

Order 637770664031457116160000: $C_2^{19}.A_{20}$

Order 510216531225165692928000: $C_2^{20}.C_2^2.A_{19}.C_2$

Order 255108265612582846464000: $C_2^{20}.C_2.A_{19}.C_2$ x 6, $C_2^{21}.A_{19}.C_2$ x 4, $(C_2^{19}\times C_4).A_{19}.C_2$ x 2, $C_2^{20}.C_2^2.A_{19}$, $C_2^{19}.C_2^2.A_{19}.C_2$, $(C_2^{18}\times C_4).C_2.A_{19}.C_2$

Order 127554132806291423232000: $C_2^{20}.A_{19}.C_2$ x 28, $(C_2^{18}\times C_4).A_{19}.C_2$ x 8, $C_2^{20}.C_2.A_{19}$ x 3, $C_2^{21}.A_{19}$ x 2, $(C_2^{19}\times C_4).A_{19}$, $(C_2^{18}\times C_4).C_2.A_{19}$

Order 80560504930289319936000: $(C_2\times A_4).C_2^{16}.C_2^3.A_{18}.C_2$

Order 63777066403145711616000: $C_2^{19}.A_{19}.C_2$ x 32, $C_2^{20}.A_{19}$ x 9, $(C_2^{18}\times C_4).A_{19}$ x 2

Order 40280252465144659968000: $(C_2\times A_4).C_2^{16}.C_2^2.A_{18}.C_2$ x 11, $(C_2^2\times S_4).C_2^{16}.C_2.A_{18}$, $(C_2^2\times S_4).C_2^{16}.A_{18}.C_2$, $(C_2^2\times A_4).C_2^{16}.C_2.A_{18}.C_2$, $(C_2\times S_4).C_2^{16}.C_2.A_{18}.C_2$

Order 31888533201572855808000: $C_2^{18}.A_{19}.C_2$ x 8, $C_2^{19}.A_{19}$ x 7

Order 26853501643429773312000: $C_2^{20}.C_2^2.A_{18}.C_2$

Order 20140126232572329984000: $(C_2\times A_4).C_2^{16}.C_2.A_{18}.C_2$ x 24, $(C_2\times A_4).C_2^{16}.C_2^2.A_{18}$ x 5, $(C_2\times S_4).C_2^{16}.A_{18}.C_2$ x 3, $(C_2^2\times S_4).C_2^{16}.A_{18}$, $(C_2^2\times S_3).C_2^{16}.C_2.A_{18}.C_2$, $(C_2^2\times A_4).C_2^{16}.A_{18}.C_2$, $(C_2\times S_4).C_2^{16}.C_2.A_{18}$

Order 17902334428953182208000: $C_2\wr A_4.C_2^{16}.C_2^2.A_{17}.C_2$

Order 15944266600786427904000: $C_2^{18}.A_{19}$

Order 13426750821714886656000: $C_2^{19}.C_2^2.A_{18}.C_2$ x 10, $C_2^{20}.C_2.A_{18}.C_2$ x 8, $(C_2\times D_4).C_2^{16}.C_2.A_{18}.C_2$ x 5, $C_2^{21}.A_{18}.C_2$ x 3, $(C_2^{19}\times C_4).A_{18}.C_2$ x 2, $C_2^{20}.C_2^2.A_{18}$, $C_2^{18}.C_2^3.A_{18}.C_2$, $C_2^2\times C_2^{19}.A_{18}.C_2$

Order 10070063116286164992000: $(C_2\times A_4).C_2^{16}.A_{18}.C_2$ x 8, $(C_2\times A_4).C_2^{16}.C_2.A_{18}$ x 5, $D_6.C_2^{16}.C_2.A_{18}.C_2$ x 4, $(C_2^{17}\times C_6).C_2.A_{18}.C_2$ x 4, $(C_2^{16}\times C_6).C_2^2.A_{18}.C_2$ x 3, $(C_2^{18}\times C_6).A_{18}.C_2$ x 2, $(C_2\times S_4).C_2^{16}.A_{18}$ x 2, $(C_2^{16}\times C_6).C_2^3.A_{18}$, $(C_2^2\times S_3).C_2^{16}.A_{18}.C_2$

Order 8951167214476591104000: $C_2^3:A_4.C_2^{16}.C_2^2.A_{17}.C_2$ x 9, $C_2^3:A_4:C_2.C_2^{16}.C_2.A_{17}.C_2$ x 3, $C_2^3:S_4.C_2^{16}.C_2.A_{17}.C_2$, $C_2\wr A_4.C_2^{16}.C_2^2.A_{17}$, $C_2\wr A_4.C_2^{16}.C_2.A_{17}.C_2$

Order 5265392479103877120000: $C_2^{21}.A_{16}.A_5.C_2^2$

Order 1974522179663953920000: $C_2^{21}.A_{15}.A_6.C_2^2$

Order 921443683843178496000: $C_2^{21}.A_{14}.A_7.C_2^2$

Order 526539247910387712000: $C_2^{21}.A_{13}.A_8.C_2^2$

Order 364527171630268416000: $C_2^{21}.A_{12}.A_9.C_2^2$

Order 303772643025223680000: $C_2^{21}.A_{11}.A_{10}.C_2^2$

Order 102181884343418880000: $C_2.S_{21}$

Order 1610915531194368000: $C_2^{21}.A_7^3.A_4.C_2^2$

Order 2958824445050880: $C_2^{14}.C_3^7.C_2^2.C_2^6.C_2^6.A_7.C_2$

Order 549755813888: $C_2^9.C_2^6.C_2^6.C_2^6.C_2^6.C_2^6$

Order 253671505920: $C_2^{20}.\PSL(3,4).D_6$

Order 10569646080: $C_2^{21}.A_7.C_2$

Order 704643072: $C_2^{21}.\SO(3,7)$

Order 2097152: $C_2^{21}$

Order 1048576: $C_2^{20}$

Order 19683: $C_3^7.C_3^2$

Order 625: $C_5^4$

Order 343: $C_7^3$

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 107145471557284795514880000: $C_2^{21}.A_{21}.C_2$ ($C_1$)

Order 53572735778642397757440000: $C_2^{20}.A_{21}.C_2$ ($C_2$) x 2, $C_2^{21}.A_{21}$ ($C_2$)

Order 26786367889321198878720000: $C_2^{20}.A_{21}$ ($C_2^2$)

Order 2097152: $C_2^{21}$ ($S_{21}$)

Order 1048576: $C_2^{20}$ ($C_2.S_{21}$)

Order 2: $C_2$ ($C_2^{20}.A_{21}.C_2$)

Order 1: $C_1$ ($C_2^{21}.A_{21}.C_2$)

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

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

The $35002 \times 35002$ rational character table is not available for this group.