Finite Group DatabasebetaFinite Group Databasebeta| order | 20096 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C64 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D157⋊C64 |
| derived, Sylow | C157 ⋊ C2×C64 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | #10597845 |
| commutator, hypocenter | C157 | C2×C64 |
| Frattini | C16 | C2×#10597845 |
| Fitting | C32×C157 | C4 |
| socle | C2×C157 | C2×C32 |
| prime radical | C2×D157 | C32 |
| cosocle | #19097677 | C22 |
| LCS | G ▷ C157 | |
| UCS | C1 ◁ C32 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 221 |
| transitive degree | μt(G) = 10048 |
A minimal faithful permutation representation — 2 generators on 221 points:
(1,2)(3,11)(4,5)(6,7)(8,24)(9,10)(12,20)(13,14)(15,16)(17,41)(18,19)(21,37)(22,23)(25,33)(26,27)(28,29)(30,62)(31,32)(34,58)(35,36)(38,54)(39,40)(42,50)(43,44)(45,46)(47,87)(48,49)(51,83)(52,53)(55,79)(56,57)(59,75)(60,61)(63,71)(64,65)(66,67)(68,116)(69,70)(72,112)(73,74)(76,108)(77,78)(80,104)(81,82)(84,100)(85,86)(88,96)(89,90)(91,92)(93,142)(94,95)(97,140)(98,99)(101,138)(102,103)(105,136)(106,107)(109,133)(110,111)(113,129)(114,115)(117,125)(118,119)(120,121)(122,139)(123,124)(126,137)(127,128)(130,134)(131,132)(141,149)(143,148)(144,145)(146,147)(150,155)(151,152)(153,154)(156,157);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,91)(53,72,73,87)(57,76,77,83)(61,80,81,79)(65,84,85,75)(67,88,89,71)(70,93,94,120)(74,97,98,116)(78,101,102,112)(82,105,106,108)(86,109,110,104)(90,113,114,100)(92,117,118,96)(95,122,123,146)(99,126,127,142)(103,130,131,140)(107,134,135,138)(111,137,132,136)(115,139,128,133)(119,141,124,129)(121,143,144,125)(145,150,151,149)(147,152,153,148)(154,156,157,155)(158,159,160,165,161,166,170,180,162,167,171,181,174,184,190,200,163,168,172,182,175,185,191,201,177,187,193,203,196,206,210,215,164,169,173,183,176,186,192,202,178,188,194,204,197,207,211,216,179,189,195,205,198,208,212,217,199,209,213,218,214,219,220,221)Polycyclic presentation:
2^7,157|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,h^a=h^129,h^b=h^156,h^c=h^1561376 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20096, 19678);