Finite Group DatabasebetaFinite Group Databasebeta| order | 30858 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C139⋊C3)⋊D37 |
| Fitting, socle, derived, Hall | C37×C139 ⋊ C2×C3 |
| Hall | C37×C139⋊C3 ⋊ C2 |
| Hall | D5143 ⋊ C3 |
| Hall | C139⋊C3 ⋊ D37 |
| Sylow | C139 ⋊ C3×D37 |
| Sylow | C37 ⋊ D139⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C37×C139 | C2×C3 |
| LCS | G ▷ C37×C139 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 176 |
| transitive degree | μt(G) = 5143 |
A minimal faithful permutation representation — 2 generators on 176 points:
(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)(39,43,44,40,41,45)(42,51,52,46,47,61)(48,64,65,54,55,78)(49,63,59,57,53,50)(56,83,84,67,68,101)(58,81,82,70,66,75)(60,62,80,72,73,76)(69,106,107,86,87,130)(71,104,105,89,85,98)(74,79,103,91,92,99)(77,119,117,95,96,94)(88,135,136,109,110,159)(90,133,134,112,108,127)(93,102,132,114,115,128)(97,146,147,120,121,123)(100,150,144,124,125,118)(111,151,149,126,138,169)(113,162,163,139,137,156)(116,131,161,141,142,157)(122,164,168,140,148,152)(129,171,166,153,154,145)(143,160,170,155,165,175)(158,176,174,172,173,167);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(38,39,42,48,56,69,88,111,124,92,55,66,76,78,98,123,149,170,176,165,110,137,148,151,154,157,159,168,134,84,103,119,104,83,102,129,155,141,114,91,72,59,44,43,49,58,71,90,113,140,120,95,73,47,53,61,75,94,118,145,167,172,142,87,108,121,125,128,130,156,163,107,132,150,146,133,106,131,158,174,166,144,117,82,52,63,51,62,77,97,122,139,112,89,70,57,41,45,50,60,74,93,116,143,153,115,68,85,96,99,101,127,152,136,161,171,138,164,162,135,160,173,175,169,147,105,65,80,81,64,79,100,126,109,86,67,54,46,40)Polycyclic presentation:
2,3,37,139|c^a=c^36,d^a=d^138,d^b=d^96911 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30858, 21);