Finite Group DatabasebetaFinite Group Databasebeta| order | 23002 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D11501 |
| unambiguous semidirect product | C7⋊D1643 |
| unambiguous semidirect product | C31⋊D371 |
| unambiguous semidirect product | C53⋊D217 |
| Fitting, socle, derived, Hall | C7×C31×C53 ⋊ C2 |
| Hall | C31×C53 ⋊ D7 |
| Hall | C7×C53 ⋊ D31 |
| Hall | C7×C31 ⋊ D53 |
| Sylow | C53 ⋊ D217 |
| Sylow | C31 ⋊ D371 |
| Sylow | C7 ⋊ D1643 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C31×C53 | C2 |
| LCS | G ▷ C7×C31×C53 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 11501 |
A minimal faithful permutation representation — 2 generators on 91 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);(2,7)(3,6)(4,5)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)Polycyclic presentation:
2,7,31,53|b^a=b^6,c^a=c^30,d^a=d^525752 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23002, 8);