Finite Group DatabasebetaFinite Group Databasebeta| order | 20096 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | #585×C157 ⋊ C2 |
| Sylow | C157 ⋊ 2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D157 |
| commutator, socle, cosocle | C2×C157 | C26 |
| Fitting | #585×C157 | C2 |
| hypercenter | #585 | D157 |
| hypocenter | C157 | 2−1+6 |
| LCS | G ▷ C2×C157 ▷ C157 | |
| UCS | C1 ◁ C2 ◁ #585 | |
5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 189 |
| transitive degree | μt(G) = 5024 |
A minimal faithful permutation representation — 6 generators on 189 points:
(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)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(159,167)(164,168)(165,179)(166,181)(176,182)(177,183)(178,180)(188,189);(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)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)(128,129)(130,131)(132,133)(134,135)(136,137)(138,139)(140,141)(142,143)(144,145)(146,147)(148,149)(150,151)(152,153)(154,155)(156,157)(160,171)(164,168)(169,185)(170,187)(176,182)(177,183)(184,186)(188,189);(158,162,163,175)(159,166,167,181)(160,170,171,187)(161,172,173,174)(164,177,168,183)(165,178,179,180)(169,184,185,186)(176,188,182,189);(158,161,163,173)(159,165,167,179)(160,169,171,185)(162,174,175,172)(164,176,168,182)(166,180,181,178)(170,186,187,184)(177,189,183,188);(158,160)(159,164)(161,169)(162,170)(163,171)(165,176)(166,177)(167,168)(172,184)(173,185)(174,186)(175,187)(178,188)(179,182)(180,189)(181,183);(158,159)(160,168)(161,165)(162,166)(163,167)(164,171)(169,182)(170,183)(172,178)(173,179)(174,180)(175,181)(176,185)(177,187)(184,189)(186,188)Polycyclic presentation:
2^7,157|c^2=g,d^2=g,b^a=bg,f^a=fg,e^b=eg,d^c=dg,h^e=h^156,h^f=h^1562717 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20096, 19323);