Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62600 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3562 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#529 ◁ G | |
1670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 832 |
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)(16,23)(17,21)(18,27)(19,22)(20,25)(24,29)(26,28)(30,31)(32,37)(33,45)(34,39)(35,49)(36,40)(38,48)(41,54)(42,51)(43,52)(44,58)(46,53)(47,59)(50,56)(55,60)(57,61);(14,16)(15,20)(17,24)(18,26)(19,25)(21,28)(22,23)(27,29)(30,32)(31,37)(33,41)(34,42)(35,44)(36,43)(38,50)(39,51)(40,52)(45,54)(46,55)(47,57)(48,56)(49,58)(53,60)(59,61);(14,17,19,18)(15,21,22,27)(16,24,25,26)(20,28,23,29)(30,33,35,48,36,47,34,46)(31,38,39,45,40,53,49,59)(32,41,44,56,43,57,42,55)(37,50,51,54,52,60,58,61);(16,25)(20,23)(24,26)(28,29)(30,34,36,35)(31,39,40,49)(32,42,43,44)(33,46,47,48)(37,51,52,58)(38,45,53,59)(41,55,57,56)(50,54,60,61);(14,18,19,17)(15,21,22,27)(16,26,25,24)(20,28,23,29)(32,43)(33,47)(37,52)(38,53)(42,44)(45,59)(46,48)(51,58)637 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55588);