Abstract group 1275541328062914232320000.b: $C_2^{19}.A_{20}.C_2$

• Label: 127...000.b
• Order: \( 2^{37} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• Exponent: \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{38} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $40$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$C_2^{19}.A_{20}.C_2$
Order:	\(127\!\cdots\!000\)\(\medspace = 2^{37} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
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 \(255\!\cdots\!000\)\(\medspace = 2^{38} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \)
Composition factors:	$C_2$ x 20, $A_{20}$
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	77	78	80	84	88	90	91	96	99	102	104	105	110	112	120	126	130	132	140	144	154	156	165	168	176	180	182	198	210	220	240	252	260	264	280	308	312	330	336	360	420	440	504	560	840
Elements	1	63003419470847	10212735104853920	17203133482517140480	10740238573479450624	565340273523554719840	141229689307545600	3462482266327540039680	506291534199108403200	5735950395546231318528	624121085952000	30268077740344121016320	152093507715072000	1878315779560406630400	10312031041947619246080	33202980806816327270400	1563163392233963520000	46196173054626752102400	33566877054287216640000	80135094764237181419520	116283039142005964800	105470846436114432000	62158629757948054732800	990280828732833792000	22677182145705856204800	46074503655081969500160	26573777667977379840000	123446157832617984000	29700104452445306880000	344376231068329574400	56949136786112643072000	33566877054287216640000	723965096723742720000	22657670098406678200320	23014621934403983769600	4778211118424260608000	600678099369905356800	36349424516670160896000	12505307137871708160000	11702682857628499968000	30197474622701568000	19835489670918045696000	84219052980331012423680	869974864130211840000	1226482046214340608000	11684884844629721088000	18757960706807562240000	16798930401611848089600	19392530607702540288000	1035341987064053760000	19376712882900172800000	12957880158099446169600	47817633303330029568000	13105703986252480512000	18045255897660889497600	7008468835510517760000	13286888833988689920000	6442127919509667840000	12505307137871708160000	8176546974762270720000	184276493947581235200	4559818668027936768000	5314755533595475968000	42907072422653145907200	9886078001479680000000	13491302508357746688000	21440206982118113280000	21132875779083023155200	8857925889325793280000	11388761857704591360000	19419299065060392960000	966319187926450176000	18394256420175347712000	7247393909448376320000	17494403631418441728000	7008468835510517760000	6442127919509667840000	8850807913164727910400	7972133300393213952000	7086340711460634624000	6959798913041694720000	4905928184857362432000	4831595939632250880000	5030036487152861184000	4141367948256215040000	4088273487381135360000	6764234315485151232000	3796253952568197120000	3543170355730317312000	10502969268772012032000	2898957563779350528000	2530835968378798080000	2277752371540918272000	1518501581027278848000	1275541328062914232320000
Conjugacy classes	1	66	6	450	4	577	2	557	5	190	1	2354	1	91	12	172	1	170	1	626	6	29	1493	19	258	597	7	3	5	3	289	1	2	393	253	62	4	260	1	28	1	152	1238	2	1	59	2	79	108	1	24	68	437	26	68	1	2	1	1	8	3	17	24	438	24	7	56	108	10	7	14	1	140	2	50	1	1	85	12	42	12	2	10	34	2	2	7	10	10	60	2	2	2	8	12484
Divisions	1	66	6	450	4	577	2	557	5	190	1	2354	1	91	12	172	1	170	1	626	6	29	1493	19	258	597	7	3	5	3	289	1	2	393	253	62	4	260	1	28	1	152	1238	2	1	59	2	79	108	1	24	68	437	26	68	1	2	1	1	8	3	17	24	438	24	7	56	108	10	7	14	1	140	2	50	1	1	85	12	42	12	2	10	34	2	2	7	10	10	60	2	2	2	8	12484
Autjugacy classes	1	50	6	355	4	440	2	446	5	145	1	1804	1	70	12	140	1	133	1	477	6	23	1152	15	196	463	6	3	4	3	212	1	2	301	197	46	4	201	1	20	1	114	921	2	1	47	1	63	80	1	20	52	321	19	56	1	1	1	1	6	3	14	18	327	21	6	39	78	7	6	9	1	104	1	33	1	1	70	8	31	8	1	7	25	1	1	7	7	7	41	1	1	1	4	9549

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 $40$ $\langle(1,20,2,19)(3,38,36,7,21,10,16,17,13,32,27,34,25,24,30)(4,37,35,8,22,9,15,18,14,31,28,33,26,23,29) \!\cdots\! \rangle$
Transitive group:	40T315783				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:	$C_2^{19}$ . $S_{20}$	$(C_2^{19}.A_{20})$ . $C_2$	$C_2$ . $(C_2^{18}.A_{20}.C_2)$		more information

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 5 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 $C_2^{19}.A_{20}$
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^6.C_2^6.C_2^5.C_2^5$
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 232050 (order at least 5496838302361190400) are shown, as well as all normal subgroups of any index.

Order 1275541328062914232320000: $C_2^{19}.A_{20}.C_2$

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

Order 63777066403145711616000: $C_2^{19}.A_{19}.C_2$

Order 31888533201572855808000: $C_2^{18}.A_{19}.C_2$ x 2, $C_2^{19}.A_{19}$

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

Order 6713375410857443328000: $C_2^{19}.C_2.A_{18}.C_2$

Order 3356687705428721664000: $C_2^{19}.A_{18}.C_2$ x 4, $(C_2^{17}\times C_4).A_{18}.C_2$ x 2, $C_2^{19}.C_2.A_{18}$

Order 1678343852714360832000: $C_2^{18}.A_{18}.C_2$ x 6, $C_2^{19}.A_{18}$ x 2, $C_2^{18}.A_{18}.C_2$ x 2, $(C_2^{17}\times C_4).A_{18}$

Order 1118895901809573888000: $A_4.C_2^{16}.C_2^2.A_{17}.C_2$

Order 839171926357180416000: $C_2^{17}.A_{18}.C_2$ x 3, $C_2^{18}.A_{18}$ x 2, $C_2^{17}.A_{18}.C_2$, $C_2^{18}.A_{18}$

Order 559447950904786944000: $S_4.C_2^{16}.A_{17}.C_2$ x 3, $A_4.C_2^{16}.C_2.A_{17}.C_2$ x 2, $A_4.C_2^{16}.C_2^2.A_{17}$, $(C_2\times A_4).C_2^{16}.A_{17}.C_2$

Order 419585963178590208000: $C_2^{17}.A_{18}$

Order 372965300603191296000: $C_2\times C_2^{18}.C_2.A_{17}.C_2$

Order 279723975452393472000: $A_4.C_2^{16}.A_{17}.C_2$ x 4, $A_4.C_2^{16}.C_2.A_{17}$ x 3, $(C_2^{15}\times C_6).C_2^2.A_{17}.C_2$

Order 263269623955193856000: $C_2^3:S_4.C_2.C_2^{14}.C_2.A_{16}.C_2$

Order 186482650301595648000: $C_2^{19}.A_{17}.C_2$ x 4, $C_2^{18}.C_2.A_{17}.C_2$ x 4, $D_4.C_2^{16}.A_{17}.C_2$ x 2, $(C_2^{17}\times C_4).A_{17}.C_2$ x 2, $D_4.C_2^{16}.C_2.A_{17}$, $C_2^{17}.C_2^2.A_{17}.C_2$, $(C_2^{16}\times C_4).C_2.A_{17}.C_2$

Order 139861987726196736000: $S_3.C_2^{16}.A_{17}.C_2$ x 2, $(C_2^{16}\times C_6).A_{17}.C_2$ x 2, $(C_2^{15}\times C_6).C_2.A_{17}.C_2$ x 2, $S_3.C_2^{16}.C_2.A_{17}$, $A_4.C_2^{16}.A_{17}$

Order 131634811977596928000: $(C_2\times C_2^3:A_4).C_2^{14}.C_2.A_{16}.C_2$ x 4, $C_2^3:S_4.C_2.C_2^{14}.A_{16}.C_2$ x 2, $C_2^3:S_4.C_2.C_2^{14}.C_2.A_{16}$

Order 93241325150797824000: $C_2^{18}.A_{17}.C_2$ x 24, $(C_2^{16}\times C_4).A_{17}.C_2$ x 8, $C_2^{19}.A_{17}$ x 2, $C_2^{18}.C_2.A_{17}$ x 2, $C_2^{17}.C_2^2.A_{17}$ x 2, $(C_2^{17}\times C_4).A_{17}$

Order 87756541318397952000: $(C_2\times Q_8:C_2^2).C_2^{14}.C_2^2.A_{16}.C_2$

Order 82271757485998080000: $C_2^{19}.A_{15}.S_5.C_2$

Order 69930993863098368000: $(C_2^{15}\times C_6).A_{17}.C_2$ x 4, $(C_2^{15}\times C_6).C_2.A_{17}$ x 2, $(C_2^{16}\times C_6).A_{17}$

Order 65817405988798464000: $(C_2\times C_2^3:A_4).C_2^{14}.A_{16}.C_2$ x 4, $(C_2\times C_2^3:A_4).C_2^{14}.C_2.A_{16}$ x 3, $(C_2\times A_4).C_2^{14}.C_2^3.A_{16}.C_2$

Order 46620662575398912000: $C_2^{17}.A_{17}.C_2$ x 21, $C_2^{18}.A_{17}$ x 7, $C_2^{17}.A_{17}.C_2$ x 2, $(C_2^{16}\times C_4).A_{17}$ x 2

Order 43878270659198976000: $(C_2^2\times D_4).C_2^{14}.C_2^2.A_{16}.C_2$ x 3, $(C_2\times C_2^2\wr C_2).C_2^{14}.C_2.A_{16}.C_2$ x 3, $C_2^{19}.C_2^2.A_{16}.C_2$ x 2, $C_2^{18}.C_2^3.A_{16}.C_2$ x 2, $(C_2^{16}\times C_4).C_2^3.A_{16}.C_2$ x 2, $(C_2\times C_2^2.D_4).C_2^{14}.C_2.A_{16}.C_2$ x 2, $(C_2^2\times D_4).C_2^{14}.C_2^3.A_{16}$

Order 41135878742999040000: $C_2^{18}.A_{15}.S_5.C_2$ x 4, $C_2^{19}.A_{15}.S_5$ x 2, $C_2^{19}.A_{15}.A_5.C_2$

Order 34965496931549184000: $(C_2^{15}\times C_6).A_{17}$

Order 32908702994399232000: $(C_2\times A_4).C_2^{14}.C_2^2.A_{16}.C_2$ x 13, $C_2^{19}.A_{14}.A_6.C_2^2$, $(C_2^2\times S_4).C_2^{14}.C_2.A_{16}$, $(C_2^2\times A_4).C_2^{14}.C_2.A_{16}.C_2$, $(C_2\times \SL(2,3)).C_2^{14}.C_2.A_{16}.C_2$, $(C_2\times S_4).C_2^{14}.C_2.A_{16}.C_2$, $(C_2\times C_2^3:A_4).C_2^{14}.A_{16}$

Order 23310331287699456000: $C_2^{16}.A_{17}.C_2$ x 6, $C_2^{17}.A_{17}$ x 5

Order 21939135329599488000: $C_2^{18}.C_2^2.A_{16}.C_2$ x 19, $(C_2^{16}\times C_4).C_2^2.A_{16}.C_2$ x 9, $(C_2^2\times D_4).C_2^{14}.C_2.A_{16}.C_2$ x 8, $C_2^{19}.C_2.A_{16}.C_2$ x 5, $C_2^{17}.C_2^3.A_{16}.C_2$ x 4, $(C_2^2\times D_4).C_2^{14}.C_2^2.A_{16}$ x 3, $C_2^{18}.C_2^3.A_{16}$ x 2, $(C_2^{17}\times C_4).C_2.A_{16}.C_2$, $(C_2\times Q_8:C_2^2).C_2^{14}.C_2.A_{16}$, $(C_2\times C_2^2\wr C_2).C_2^{14}.C_2.A_{16}$, $(C_2\times C_2^2:C_4).C_2^{14}.C_2.A_{16}.C_2$, $(C_2\times C_2^2.D_4).C_2^{14}.A_{16}.C_2$

Order 20567939371499520000: $C_2^{18}.A_{15}.S_5$ x 4, $C_2^{18}.A_{15}.A_5.C_2$ x 2, $C_2^{19}.A_{15}.A_5$

Order 16454351497199616000: $(C_2\times A_4).C_2^{14}.C_2.A_{16}.C_2$ x 25, $(C_2\times S_4).C_2^{14}.A_{16}.C_2$ x 5, $(C_2\times A_4).C_2^{14}.C_2^2.A_{16}$ x 5, $C_2^{18}.A_{14}.A_6.C_2^2$ x 4, $(C_2^2\times A_4).C_2^{14}.A_{16}.C_2$ x 4, $C_2^{19}.A_{14}.A_6.C_2$ x 3, $(C_2\times \SL(2,3)).C_2^{14}.A_{16}.C_2$ x 2, $(C_2\times S_4).C_2^{14}.C_2.A_{16}$ x 2, $D_6.C_2^{14}.C_2^2.A_{16}.C_2$, $C_2^{19}.A_{13}.A_7.C_2^2$, $C_2^3:A_4.C_2^{14}.C_2^3.A_{15}.C_2$, $(C_2^2\times S_4).C_2^{14}.A_{16}$, $(C_2\times \SL(2,3)).C_2^{14}.C_2.A_{16}$

Order 13807847410237440000: $C_2^{19}.A_{10}^2.D_4$

Order 13711959580999680000: $C_2^4:D_5.C_2^{14}.C_2^2.A_{15}.C_2$

Order 11655165643849728000: $C_2^{16}.A_{17}$

Order 10969567664799744000: $C_2^{18}.C_2.A_{16}.C_2$ x 46, $C_2^{17}.C_2^2.A_{16}.C_2$ x 26, $(C_2^{16}\times C_4).C_2.A_{16}.C_2$ x 12, $C_2^{19}.A_{16}.C_2$ x 11, $(C_2^{17}\times C_4).A_{16}.C_2$ x 8, $C_2^{16}.C_2^3.A_{16}.C_2$ x 6, $(C_2^2\times D_4).C_2^{14}.A_{16}.C_2$ x 6, $C_2^{18}.C_2^2.A_{16}$ x 4, $(C_2^{15}\times C_4^2).A_{16}.C_2$ x 4, $(C_2^2\times D_4).C_2^{14}.C_2.A_{16}$ x 4, $C_2^{17}.C_2^3.A_{16}$ x 3, $(C_2^{16}\times C_4).C_2^2.A_{16}$ x 3, $(C_2^{15}\times C_4).C_2^2.A_{16}.C_2$ x 3, $(C_2\times D_4).C_2^{14}.C_2.A_{16}.C_2$ x 2, $(C_2\times C_2^2:C_4).C_2^{14}.A_{16}.C_2$ x 2, $(C_2^{15}\times C_8).C_2.A_{16}.C_2$, $(C_2\times D_4:C_2).C_2^{14}.C_2.A_{16}$, $(C_2\times C_2^2\wr C_2).C_2^{14}.A_{16}$, $(C_2\times C_2^2.D_4).C_2^{14}.A_{16}$

Order 10283969685749760000: $C_2^{18}.A_{15}.A_5$

Order 10125754767507456000: $C_2^{19}.A_{12}.A_8.C_2^2$

Order 8227175748599808000: $(C_2\times A_4).C_2^{14}.A_{16}.C_2$ x 12, $C_2^3:A_4.C_2^{14}.C_2^2.A_{15}.C_2$ x 9, $(C_2^{14}\times C_6).C_2^2.A_{16}.C_2$ x 7, $C_2^{18}.A_{14}.A_6.C_2$ x 6, $(C_2\times A_4).C_2^{14}.C_2.A_{16}$ x 6, $C_2^{18}.A_{13}.A_7.C_2^2$ x 4, $D_6.C_2^{14}.C_2.A_{16}.C_2$ x 3, $C_2^{19}.A_{13}.A_7.C_2$ x 3, $(C_2\times S_4).C_2^{14}.A_{16}$ x 3, $C_2^3:S_4.C_2^{14}.C_2.A_{15}.C_2$ x 2, $C_2^3:A_4:C_2.C_2^{14}.C_2.A_{15}.C_2$ x 2, $(C_2^{16}\times C_6).A_{16}.C_2$ x 2, $\GL(2,\mathbb{Z}/4).C_2^{14}.C_2^2.A_{15}.C_2$, $C_2^{19}.A_{14}.A_6$, $C_2^3:A_4.C_2^{14}.C_2^3.A_{15}$, $C_2\wr A_4.C_2^{14}.C_2.A_{15}.C_2$, $(C_2^{15}\times C_6).C_2.A_{16}.C_2$, $(C_2^{14}\times C_6).C_2^3.A_{16}$, $(C_2^2\times S_3).C_2^{14}.A_{16}.C_2$, $(C_2^2\times A_4).C_2^{14}.A_{16}$, $(C_2\times \SL(2,3)).C_2^{14}.A_{16}$

Order 7594316075630592000: $C_2^{19}.A_{11}.A_9.C_2^2$

Order 6903923705118720000: $C_2^{18}.A_{10}^2.D_4$ x 4, $C_2^{19}.A_{10}^2.C_4$, $C_2^{19}.A_{10}^2.C_2^2$, $C_2^{19}.A_{10}.A_{10}.C_2^2$

Order 6855979790499840000: $C_2^4:D_5.C_2^{14}.C_2.A_{15}.C_2$ x 3, $C_2^4:C_5:C_4.C_2^{14}.C_2.A_{15}$, $C_2^4:C_5:C_4.C_2^{14}.A_{15}.C_2$, $C_2^4:C_5.C_2^{14}.C_2^2.A_{15}.C_2$, $(C_2\times C_2^4:D_5).C_2^{14}.A_{15}.C_2$

Order 4865804016353280000: $C_2.S_{20}$ x 2

Order 2609192632320000: $C_2^{17}.A_5^4.C_2\wr C_2^2.S_4$

Order 1948197165465600: $C_2^{20}.C_2.C_2^8.A_{10}.C_2$

Order 500964985405440: $C_2^{15}.C_2^{10}.C_3^5.C_2^5.C_2^4.S_5$

Order 137438953472: $C_2^9.C_2^6.C_2^6.C_2^6.C_2^5.C_2^5$

Order 3586129920: $C_2^{19}.\PSL(2,19).C_2$

Order 524288: $C_2^{19}$

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 1275541328062914232320000: $C_2^{19}.A_{20}.C_2$ ($C_1$)

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

Order 524288: $C_2^{19}$ ($S_{20}$)

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

Order 1: $C_1$ ($C_2^{19}.A_{20}.C_2$)

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 $12484 \times 12484$ rational character table is not available for this group.