Finite Group DatabasebetaFinite Group Databasebeta| order | 19558 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D9779 |
| unambiguous semidirect product | C7⋊D1397 |
| unambiguous semidirect product | C11⋊D889 |
| unambiguous semidirect product | C127⋊D77 |
| Fitting, socle, derived, Hall | C7×C11×C127 ⋊ C2 |
| Hall | C11×C127 ⋊ D7 |
| Hall | C7×C127 ⋊ D11 |
| Sylow | C127 ⋊ D77 |
| Hall | C7×C11 ⋊ D127 |
| Sylow | C11 ⋊ D889 |
| Sylow | C7 ⋊ D1397 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C11×C127 | C2 |
| LCS | G ▷ C7×C11×C127 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 9779 |
A minimal faithful permutation representation — 2 generators on 145 points:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(2,7)(3,6)(4,5)(9,18)(10,17)(11,16)(12,15)(13,14)(20,145)(21,144)(22,143)(23,142)(24,141)(25,140)(26,139)(27,138)(28,137)(29,136)(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:
2,7,11,127|b^a=b^6,c^a=c^10,d^a=d^1264891 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19558, 9);