Finite Group DatabasebetaFinite Group Databasebeta| order | 32370 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D16185 |
| unambiguous semidirect product | C3⋊D5395 |
| unambiguous semidirect product | C5⋊D3237 |
| unambiguous semidirect product | C13⋊D1245 |
| unambiguous semidirect product | C83⋊D195 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C5×C13×C83 | C2 |
| LCS | G ▷ C3×C5×C13×C83 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 104 |
| transitive degree | μt(G) = 16185 |
A minimal faithful permutation representation — 2 generators on 104 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);(2,3)(5,8)(6,7)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(23,104)(24,103)(25,102)(26,101)(27,100)(28,99)(29,98)(30,97)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)Polycyclic presentation:
2,3,5,13,83|b^a=b^2,c^a=c^4,d^a=d^12,e^a=e^828094 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×D16185 |
| maximal normal subgroup of | C2×D16185 |
| maximal quotient of | Dic16185, C2×D16185 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32370, 17);