Finite Group DatabasebetaFinite Group Databasebeta| order | 13188 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #20268646⋊C2 |
| unambiguous semidirect product | #20268650⋊C2 |
| unambiguous semidirect product | #20268654⋊C2 |
| Fitting, socle, derived, Hall | C7×C157 ⋊ C22×C3 |
| Hall | D7×D157 ⋊ C3 |
| Hall | #15693074 ⋊ C22 |
| Sylow | C157 ⋊ C2×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C2×D157⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C157 | C22×C3 |
| LCS | G ▷ C7×C157 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 164 |
| transitive degree | μt(G) = 1099 |
A minimal faithful permutation representation — 2 generators on 164 points:
(1,2,8)(3,12,24)(4,6,13)(5,16,7)(9,25,41)(10,26,23)(11,29,15)(14,38,20)(17,42,64)(18,43,37)(19,48,28)(21,39,22)(27,61,33)(30,65,93)(31,66,56)(32,71,47)(34,59,45)(35,40,62)(36,44,57)(46,90,52)(49,94,123)(50,95,79)(51,100,70)(53,84,68)(54,63,91)(55,82,81)(58,67,86)(60,88,80)(69,120,75)(72,73,124)(74,122,99)(76,109,97)(77,92,121)(78,112,105)(83,96,116)(85,114,107)(87,106,111)(89,118,104)(98,148,101)(102,132,126)(103,141,127)(108,125,145)(110,138,130)(113,129,140)(115,143,134)(117,136,135)(119,147,128)(131,146,154)(133,144,150)(137,149,157)(139,156,152)(142,151,153)(159,162,163,160,161,164);(2,3)(4,6)(5,9)(7,8)(10,14)(11,17)(12,20)(13,22)(15,24)(16,26)(18,27)(19,30)(21,33)(23,35)(25,36)(28,41)(29,44)(31,46)(32,49)(34,52)(37,54)(38,45)(39,43)(40,58)(42,55)(47,64)(48,67)(50,69)(51,72)(53,75)(56,77)(57,80)(59,82)(60,83)(61,68)(62,66)(63,87)(65,78)(70,93)(71,96)(73,98)(74,79)(76,101)(81,104)(84,106)(85,108)(86,107)(88,112)(89,113)(90,97)(91,95)(92,117)(94,103)(99,123)(100,125)(102,127)(105,128)(109,129)(110,131)(111,134)(114,136)(115,137)(116,130)(118,141)(119,142)(120,126)(121,124)(122,146)(132,149)(133,135)(138,151)(139,153)(140,152)(143,154)(144,155)(145,150)(147,148)(156,157)(158,159,161,164,163,162,160)Polycyclic presentation:
2^2,3,7,157|e^a=e^156,d^b=d^6,d^c=d^2,e^c=e^144144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13188, 30);