Finite Group DatabasebetaFinite Group Databasebeta| order | 23690 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D11845 |
| unambiguous semidirect product | C5⋊D2369 |
| unambiguous semidirect product | C23⋊D515 |
| unambiguous semidirect product | C103⋊D115 |
| Fitting, socle, derived, Hall | C5×C23×C103 ⋊ C2 |
| Hall | C23×C103 ⋊ D5 |
| Hall | C5×C103 ⋊ D23 |
| Hall | C5×C23 ⋊ D103 |
| Sylow | C103 ⋊ D115 |
| Sylow | C23 ⋊ D515 |
| Sylow | C5 ⋊ D2369 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C23×C103 | C2 |
| LCS | G ▷ C5×C23×C103 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 11845 |
A minimal faithful permutation representation — 2 generators on 131 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);(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)(30,131)(31,130)(32,129)(33,128)(34,127)(35,126)(36,125)(37,124)(38,123)(39,122)(40,121)(41,120)(42,119)(43,118)(44,117)(45,116)(46,115)(47,114)(48,113)(49,112)(50,111)(51,110)(52,109)(53,108)(54,107)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)Polycyclic presentation:
2,5,23,103|b^a=b^4,c^a=c^22,d^a=d^1025924 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23690, 8);