Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62161 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#529 ◁ G | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 5 generators on 61 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,21)(16,22,23,19)(17,25,26,20)(24,27,29,28)(30,31)(32,37)(33,45)(34,39)(35,48)(36,40)(38,46)(41,55)(42,51)(43,52)(44,58)(47,53)(49,54)(50,56)(57,60)(59,61);(14,16)(15,19)(17,24)(18,23)(20,27)(21,22)(25,28)(26,29)(30,32)(31,37)(33,41)(34,42)(35,44)(36,43)(38,50)(39,51)(40,52)(45,55)(46,56)(47,57)(48,58)(49,59)(53,60)(54,61);(30,33,36,47)(31,38,40,54)(32,41,43,57)(34,46,35,49)(37,50,52,61)(39,53,48,45)(42,56,44,59)(51,60,58,55);(16,23)(19,22)(24,29)(27,28)(30,34,36,35)(31,39,40,48)(32,42,43,44)(33,46,47,49)(37,51,52,58)(38,53,54,45)(41,56,57,59)(50,60,61,55);(14,17,18,26)(15,20,21,25)(16,24,23,29)(19,27,22,28)(32,43)(37,52)(41,57)(42,44)(50,61)(51,58)(55,60)(56,59)637 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55149);