Finite Group DatabasebetaFinite Group Databasebeta| order | 37698 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C61⋊C3)⋊D103 |
| Fitting, socle, derived, Hall | C61×C103 ⋊ C2×C3 |
| Hall | C103×C61⋊C3 ⋊ C2 |
| Hall | D6283 ⋊ C3 |
| Hall | C61⋊C3 ⋊ D103 |
| Sylow | C103 ⋊ D61⋊C3 |
| Sylow | C61 ⋊ C3×D103 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C61×C103 | C2×C3 |
| LCS | G ▷ C61×C103 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 164 |
| transitive degree | μt(G) = 6283 |
A minimal faithful permutation representation — 2 generators on 164 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,40)(12,26,22,20,16,13)(19,45,46,30,31,56)(21,43,44,32,29,37)(23,25,42,34,35,38)(33,58,59,48,47,54)(36,41,57,50,49,55)(39,61,60,52,53,51)(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)(157,158)(159,160)(161,162)(163,164);(1,2,5,11,19,31,47,53,55,56,50,34,22,7,6,12,21,33,49,18,29,38,40,54,60,44,15,26,14,25,39,52,35,10,16,24,37,51,59,28,42,43,27,41,48,32,20,4,8,13,23,36,46,57,61,58,45,30,17,9,3)(62,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131,133,135,137,139,141,143,145,147,149,151,153,155,157,159,161,163,164,162,160,158,156,154,152,150,148,146,144,142,140,138,136,134,132,130,128,126,124,122,120,118,116,114,112,110,108,106,104,102,100,98,96,94,92,90,88,86,84,82,80,78,76,74,72,70,68,66,64)Polycyclic presentation:
2,3,61,103|c^a=c^60,d^a=d^102,c^b=c^471189 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(37698, 22);