Finite Group DatabasebetaFinite Group Databasebeta| order | 24610 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D12305 |
| unambiguous semidirect product | C5⋊D2461 |
| unambiguous semidirect product | C23⋊D535 |
| unambiguous semidirect product | C107⋊D115 |
| Fitting, socle, derived, Hall | C5×C23×C107 ⋊ C2 |
| Hall | C23×C107 ⋊ D5 |
| Hall | C5×C107 ⋊ D23 |
| Hall | C5×C23 ⋊ D107 |
| Sylow | C107 ⋊ D115 |
| Sylow | C23 ⋊ D535 |
| Sylow | C5 ⋊ D2461 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C23×C107 | C2 |
| LCS | G ▷ C5×C23×C107 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 12305 |
A minimal faithful permutation representation — 2 generators on 135 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(2,5)(3,4)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(30,135)(31,134)(32,133)(33,132)(34,131)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,124)(42,123)(43,122)(44,121)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)Polycyclic presentation:
2,5,23,107|b^a=b^4,c^a=c^22,d^a=d^1066154 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24610, 8);