Finite Group DatabasebetaFinite Group Databasebeta| order | 24186 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C29⋊(D139⋊C3) |
| unambiguous semidirect product | (C139⋊C3)⋊D29 |
| unambiguous semidirect product | D4031⋊C3 |
| Fitting, socle, derived, Hall | C29×C139 ⋊ C2×C3 |
| Hall | C29×C139⋊C3 ⋊ C2 |
| Hall | D4031 ⋊ C3 |
| Hall | C139⋊C3 ⋊ D29 |
| Sylow | C139 ⋊ C3×D29 |
| Sylow | C29 ⋊ D139⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C29×C139 | C2×C3 |
| LCS | G ▷ C29×C139 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 168 |
| transitive degree | μt(G) = 4031 |
A minimal faithful permutation representation — 2 generators on 168 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,122)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,109,110,83,84,86)(63,113,107,87,88,81)(74,114,112,89,101,132)(76,125,126,102,100,119)(79,94,124,104,105,120)(85,127,131,103,111,115)(92,134,129,116,117,108)(106,123,133,118,128,138)(121,139,137,135,136,130)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168);(1,2,5,11,19,32,51,74,87,55,18,29,39,41,61,86,112,133,139,128,73,100,111,114,117,120,122,131,97,47,66,82,67,46,65,92,118,104,77,54,35,22,7,6,12,21,34,53,76,103,83,58,36,10,16,24,38,57,81,108,130,135,105,50,71,84,88,91,93,119,126,70,95,113,109,96,69,94,121,137,129,107,80,45,15,26,14,25,40,60,85,102,75,52,33,20,4,8,13,23,37,56,79,106,116,78,31,48,59,62,64,90,115,99,124,134,101,127,125,98,123,136,138,132,110,68,28,43,44,27,42,63,89,72,49,30,17,9,3)(140,141,143,145,147,149,151,153,155,157,159,161,163,165,167,168,166,164,162,160,158,156,154,152,150,148,146,144,142)Polycyclic presentation:
2,3,29,139|c^a=c^28,d^a=d^138,d^b=d^96715 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24186, 12);