Abstract group 304888344611713860501504000000.a: $S_{28}$

• Label: 304...000.a
• Order: \( 2^{25} \cdot 3^{13} \cdot 5^{6} \cdot 7^{4} \cdot 11^{2} \cdot 13^{2} \cdot 17 \cdot 19 \cdot 23 \)
• Exponent: \( 2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{25} \cdot 3^{13} \cdot 5^{6} \cdot 7^{4} \cdot 11^{2} \cdot 13^{2} \cdot 17 \cdot 19 \cdot 23 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $28$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$S_{28}$
Order:	\(304\!\cdots\!000\)\(\medspace = 2^{25} \cdot 3^{13} \cdot 5^{6} \cdot 7^{4} \cdot 11^{2} \cdot 13^{2} \cdot 17 \cdot 19 \cdot 23 \)
Exponent:	\(80313433200\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \)
Automorphism group:	Group of order \(304\!\cdots\!000\)\(\medspace = 2^{25} \cdot 3^{13} \cdot 5^{6} \cdot 7^{4} \cdot 11^{2} \cdot 13^{2} \cdot 17 \cdot 19 \cdot 23 \)
Composition factors:	$C_2$, $A_{28}$
Derived length:	$1$

This group is nonabelian, almost simple, 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	23	24	25	26	27	28	30	33	34	35	36	38	39	40	42	44	45	46	48	50	51	52	55	56	57	60	63	65	66	68	69	70	72	75	76	77	78	80	84	85	88	90	91	92	95	99	102	104	105	110	112	114	115	117	119	120	126	130	132	133	136	138	140	143	144	152	153	154	156	165	168	170	171	176	180	182	187	190	195	198	204	210	220	228	231	234	238	240	252	255	260	264	266	273	280	285	286	306	308	312	315	330	336	340	357	360	364	380	385	390	396	408	420	429	440	455	462	468	476	495	504	510	520	546	560	572	585	616	630	660	693	728	770	780	792	840	910	924	990	1092	1155	1260	1320	1365	1540	2310
Elements	1	17411277367391103	96357939154884468152	110151147765102551452800	136010110548770728530624	21718504084237946705046600	5320390302735944670748800	120508437382644361597747200	85920144282136169957952000	272190814858297825856638272	1749818323147908633984000	2515328971211736971760268800	451018261277866600851456000	1150680516032629014569136000	344747955105648053798686848	1981546115513280599162880000	449299756157091840000	5286886473200573069180352000	44220554948092723200000	4770473802839637972426547200	3096452174187759786374976000	1594100996884892489745024000	110466791525983282790400000	15490398926557104522964992000	2032588964078092403343360000	12177678671423015955240960000	11292160911544957796352000000	11598114337612546133085696000	6102334934379484687565078400	140583822036434887219200000	16037754796027393228800000	20393469642765307139251200	5447923066668161928296448000	115813633409054842060800000	3019675498134619484160000	3242977528776001774210252800	7196724118623588309528768000	3806644138161062767994880000	980233560546181063170048000	2761669788149582069760000000	5711055191176209806131200000	6097766892234277210030080000	63805058371868612198400000	79697067239302585712640000	270561920166876355670016000	1326332738638082988933120000	255064160940598827417600000	16716042642336549116135731200	202770440177903057940480000	35180716979111620460544000	6916788038892221319997440000	580028013208559281766400000	2209335830519665655808000000	2322404775961063254324864000	3676353020578325214167040000	4065177928156184806686720000	1370660321171082048307200000	47139054145374702551040000	693193283918051463106560000	3097278421452331281285120000	9150151815923030006814720000	363674005827697905500160000	537492805293167397273600000	6891517622499245571483648000	239398907865195914035200000	3314003745779498483712000000	133722958163032394956800000	94177721888088020582400000	2020065182674969625395200000	623550643741905636556800000	2399934006835791987585024000	3499148814845175458942976000	1270368102548807752089600000	3677381349483390861312000000	2651202996623598786969600000	293161869818955635097600000	106753622062925021184000000	5483051579966682941227008000	4447077537896097201377280000	1087200100949622971351040000	2503243338578989275709440000	1146196784254563385344000000	1494550708880950296576000000	2209335830519665655808000000	3767621600893708095842304000	88836930248168374272000000	2117280170914679586816000000	2005844372445485924352000000	996367139253966864384000000	1471637384102130377932800000	2920696221117377448345600000	971387570477486415052800000	2669413387366999338024960000	2167098527877377930035200000	1782972775507098599424000000	1732320139839283298304000000	3362812240982440541847552000	1738612755731861891481600000	1630418955142854868992000000	1203506623467291554611200000	1681828800964901040291840000	870972070308084102758400000	1805915439897814941696000000	6082781730162941546975232000	667218225288892210053120000	1337229581630323949568000000	128145425310358673817600000	1140073938184827469824000000	960782598566325190656000000	1693824136731743669452800000	1599842823022775749017600000	398546855701586745753600000	973948878620752609935360000	1137770758511726808268800000	1146196784254563385344000000	102373986285984507494400000	1293917443225307759738880000	1069783665304259159654400000	799532372233515368448000000	996367139253966864384000000	228226304137556371046400000	1302941643639802822656000000	184937941459486298603520000	2580164055899424941506560000	907405787534862680064000000	896730425328570177945600000	854028976503400169472000000	1185676895712220568616960000	453702893767431340032000000	802337748978194369740800000	85791092639659744296960000	1615647638113355500093440000	673680054381943504896000000	747275354440475148288000000	2678151369024742460129280000	710695441985346994176000000	461952037290475546214400000	111680712311983099084800000	1297865247625621772697600000	651470821819901411328000000	640521732377550127104000000	307968024860317030809600000	604937191689908453376000000	1195640567104760237260800000	586323739637911270195200000	884138972469866201088000000	544443472520917608038400000	533021581489010245632000000	521176657455921129062400000	494948611382652370944000000	1342960565551596766494720000	1204924897265990383042560000	439954321229024329728000000	418802671169936621568000000	560941759567006020403200000	521176657455921129062400000	384960031075396288512000000	629134679357504791511040000	335042136935949297254400000	439954321229024329728000000	307968024860317030809600000	279201780779957747712000000	131986296368707298918400000	241974876675963381350400000	230976018645237773107200000	223361424623966198169600000	197979444553060948377600000	131986296368707298918400000	304888344611713860501504000000
Conjugacy classes	1	14	9	49	5	156	4	56	12	62	2	345	2	34	30	20	1	96	1	111	17	15	1	188	1	10	1	54	294	7	5	7	107	4	5	66	138	20	16	2	30	1	3	12	4	32	3	321	9	3	46	6	1	51	40	1	4	2	29	10	142	2	10	76	2	1	1	3	12	6	19	17	4	7	1	3	1	107	34	11	41	1	2	1	45	1	3	1	1	8	24	7	46	4	1	1	47	5	1	2	5	9	6	93	13	2	4	5	2	6	20	2	8	12	1	2	16	1	2	1	6	6	5	25	2	2	1	12	4	1	2	12	5	1	51	1	4	1	10	2	1	2	5	2	2	5	1	1	1	2	10	11	1	1	3	4	1	10	1	4	1	1	1	3	1	1	1	1	3718
Divisions	1	14	9	49	5	156	4	56	12	62	2	345	2	34	30	20	1	96	1	111	17	15	1	188	1	10	1	54	294	7	5	7	107	4	5	66	138	20	16	2	30	1	3	12	4	32	3	321	9	3	46	6	1	51	40	1	4	2	29	10	142	2	10	76	2	1	1	3	12	6	19	17	4	7	1	3	1	107	34	11	41	1	2	1	45	1	3	1	1	8	24	7	46	4	1	1	47	5	1	2	5	9	6	93	13	2	4	5	2	6	20	2	8	12	1	2	16	1	2	1	6	6	5	25	2	2	1	12	4	1	2	12	5	1	51	1	4	1	10	2	1	2	5	2	2	5	1	1	1	2	10	11	1	1	3	4	1	10	1	4	1	1	1	3	1	1	1	1	3718
Autjugacy classes	1	14	9	49	5	156	4	56	12	62	2	345	2	34	30	20	1	96	1	111	17	15	1	188	1	10	1	54	294	7	5	7	107	4	5	66	138	20	16	2	30	1	3	12	4	32	3	321	9	3	46	6	1	51	40	1	4	2	29	10	142	2	10	76	2	1	1	3	12	6	19	17	4	7	1	3	1	107	34	11	41	1	2	1	45	1	3	1	1	8	24	7	46	4	1	1	47	5	1	2	5	9	6	93	13	2	4	5	2	6	20	2	8	12	1	2	16	1	2	1	6	6	5	25	2	2	1	12	4	1	2	12	5	1	51	1	4	1	10	2	1	2	5	2	2	5	1	1	1	2	10	11	1	1	3	4	1	10	1	4	1	1	1	3	1	1	1	1	3718

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Permutation group:	Degree $28$ $\langle(1,2), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)\rangle$
Transitive group:	28T1854				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:	$A_{28}$ . $C_2$				more information

Elements of the group are displayed as permutations of degree 28.

Homology

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

Subgroups

There are 3 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 $A_{28}$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_2^5.C_2^4.C_2\times D_4\times D_4^2.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^8.C_3^4.C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5\wr C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^4$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^2$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}^2$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}$
23-Sylow subgroup:	$P_{ 23 } \simeq$ $C_{23}$

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 17705952 (order at least 17219539768983552000000) are shown, as well as all normal subgroups of any index.

Order 304888344611713860501504000000: $S_{28}$

Order 152444172305856930250752000000: $A_{28}$

Order 10888869450418352160768000000: $S_{27}$

Order 5444434725209176080384000000: $A_{27}$

Order 806582922253211271168000000: $C_2\times S_{26}$

Order 403291461126605635584000000: $S_{26}$ x 2, $C_2.A_{26}$

Order 201645730563302817792000000: $A_{26}$

Order 93067260259985915904000000: $S_3\times S_{25}$

Order 46533630129992957952000000: $C_3.A_{25}.C_2$ x 2, $S_3.A_{25}$

Order 31022420086661971968000000: $C_2.A_{25}.C_2$

Order 23266815064996478976000000: $C_3.A_{25}$

Order 15511210043330985984000000: $S_{25}$, $C_2.A_{25}$, $S_{25}$

Order 14890761641597746544640000: $S_4\times S_{24}$

Order 7755605021665492992000000: $A_{25}$

Order 7445380820798873272320000: $A_4.A_{24}.C_2$ x 2, $S_4.A_{24}$

Order 4963587213865915514880000: $D_4.A_{24}.C_2$

Order 3722690410399436636160000: $S_3.A_{24}.C_2$, $A_4.A_{24}$

Order 3102242008666197196800000: $S_5\times S_{23}$

Order 2481793606932957757440000: $C_2^2.A_{24}.C_2$ x 4, $C_4.A_{24}.C_2$ x 2, $D_4.A_{24}$

Order 1861345205199718318080000: $C_3.A_{24}.C_2$ x 2, $S_3.A_{24}$

Order 1551121004333098598400000: $A_{23}.S_5$ x 2, $A_{23}.A_5.C_2$

Order 1240896803466478878720000: $C_2.A_{24}.C_2$ x 6, $C_2^2.A_{24}$ x 2, $C_4.A_{24}$

Order 930672602599859159040000: $C_3.A_{24}$

Order 809280523999877529600000: $S_6\times S_{22}$

Order 775560502166549299200000: $A_{23}.A_5$

Order 620448401733239439360000: $C_2.A_{24}$ x 2, $S_{24}$ x 2, $S_{24}$, $S_4.A_{23}.C_2$

Order 517040334777699532800000: $S_{23}\times F_5$

Order 404640261999938764800000: $A_{22}.A_6.C_2$ x 3

Order 310224200866619719680000: $A_4.A_{23}.C_2$ x 2, $S_4.A_{23}$, $D_6.A_{23}.C_2$, $A_{24}$

Order 258520167388849766400000: $D_5.A_{23}.C_2$ x 2, $A_{23}\times F_5$

Order 257498348545415577600000: $S_7\times S_{21}$

Order 206816133911079813120000: $D_4.A_{23}.C_2$

Order 202320130999969382400000: $A_{22}.A_6$

Order 155112100433309859840000: $S_3.A_{23}.C_2$ x 4, $C_6.A_{23}.C_2$ x 2, $D_6.A_{23}$, $A_4.A_{23}$

Order 134880087333312921600000: $A_{22}.S_5.C_2$, $A_5.A_{22}.C_2^2$

Order 129260083694424883200000: $C_5.A_{23}.C_2$ x 2, $D_5.A_{23}$

Order 128749174272707788800000: $A_{21}.A_7.C_2$ x 3

Order 103408066955539906560000: $C_2^2.A_{23}.C_2$ x 3, $C_4.A_{23}.C_2$ x 2, $D_4.A_{23}$, $C_2^2\times S_{23}$

Order 98094608969682124800000: $S_8\times S_{20}$

Order 80928052399987752960000: $S_3\wr C_2.A_{22}.C_2$

Order 77556050216654929920000: $C_3.A_{23}.C_2$ x 4, $S_3.A_{23}$ x 2, $C_6.A_{23}$

Order 67440043666656460800000: $A_5.A_{22}.C_2$ x 3, $A_{22}.S_5$ x 2, $A_{22}.A_5.C_2$

Order 64630041847212441600000: $C_5.A_{23}$

Order 64374587136353894400000: $A_{21}.A_7$

Order 53952034933325168640000: $(C_2\times S_4).A_{22}.C_2$ x 2

Order 51704033477769953280000: $C_2.S_{23}$ x 5, $C_2^2.A_{23}$ x 2, $C_4.A_{23}$, $C_2.A_{23}.C_2$

Order 49047304484841062400000: $A_{20}.A_8.C_2$ x 3

Order 44142574036356956160000: $S_9\times S_{19}$

Order 40464026199993876480000: $S_3^2.A_{22}.C_2$ x 4, $C_3:S_3.C_2.A_{22}.C_2$ x 2, $S_3\wr C_2\times A_{22}$

Order 38778025108327464960000: $C_3.A_{23}$

Order 36785478363630796800000: $A_6.A_{21}.C_2^2$

Order 33720021833328230400000: $A_{22}.A_5$, $A_5.A_{22}$

Order 26976017466662584320000: $S_4.A_{22}.C_2$ x 6, $(C_2\times A_4).A_{22}.C_2$ x 3, $S_{22}\times S_4$ x 2, $S_{22}\times C_2\times A_4$, $C_2\times S_4\times A_{22}$, $C_2\times S_4.A_{22}$

Order 25852016738884976640000: $C_2.A_{23}$ x 2, $S_{23}$ x 2, $S_{23}$

Order 24523652242420531200000: $A_{20}.A_8$

Order 23232933703345766400000: $S_{10}\times S_{18}$

Order 22480014555552153600000: $F_5.A_{22}.C_2$

Order 22071287018178478080000: $A_{19}.A_9.C_2$ x 3

Order 20232013099996938240000: $C_3:S_3.A_{22}.C_2$ x 4, $(C_3\times S_3).A_{22}.C_2$ x 4, $S_3^2.A_{22}$ x 2, $C_3:S_3.C_2.A_{22}$

Order 18392739181815398400000: $A_6.A_{21}.C_2$ x 3

Order 17984011644441722880000: $(C_2\times D_4).A_{22}.C_2$

Order 15200108913103994880000: $S_{14}\wr C_2$

Order 14197903929822412800000: $S_{11}\times S_{17}$

Order 10022049832815820800000: $S_{16}\times S_{12}$

Order 8142915489162854400000: $S_{13}\times S_{15}$

Order 15485790781440000: $A_7^4.C_2\wr A_4.C_2$

Order 1428329123020800: $C_2^{14}.A_{14}.C_2$

Order 23115815976960: $C_2^{14}.C_3^7.C_2^6.A_7.C_2^2$

Order 33554432: $C_2^5.C_2^5.C_2^4.C_2\times D_4\times D_4^2.C_2$

Order 1594323: $C_3^8.C_3^4.C_3$

Order 58968: $\PSL(2,27).C_6$

Order 15625: $C_5\wr C_5$

Order 2401: $C_7^4$

Order 169: $C_{13}^2$

Order 121: $C_{11}^2$

Order 23: $C_{23}$

Order 19: $C_{19}$

Order 17: $C_{17}$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 304888344611713860501504000000: $S_{28}$ ($C_1$)

Order 152444172305856930250752000000: $A_{28}$ ($C_2$)

Order 1: $C_1$ ($S_{28}$)

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 $3718 \times 3718$ rational character table is not available for this group.