Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C29 |
| direct product | C13×#62635 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C25×C13 |
| socle | C2×C13 | C23×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C13×#41 ◁ C2×C13×#131 ◁ G | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 5 generators on 77 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,27)(17,23)(18,24)(19,37)(20,41)(21,26)(22,30)(25,39)(28,52)(29,53)(31,54)(32,47)(33,48)(34,61)(35,64)(36,51)(38,44)(40,66)(42,50)(43,57)(45,67)(46,58)(49,63)(55,73)(56,74)(59,76)(60,71)(62,69)(65,72)(68,75)(70,77);(14,16)(15,22)(17,28)(18,29)(19,38)(20,42)(21,31)(23,43)(24,44)(25,50)(26,46)(27,37)(30,39)(32,55)(33,56)(34,62)(35,65)(36,58)(40,67)(41,45)(47,68)(48,69)(49,72)(51,53)(52,61)(54,66)(57,63)(59,77)(60,75)(64,70)(71,74)(73,76);(14,17)(15,23)(16,28)(18,33)(19,34)(20,35)(21,32)(22,43)(24,48)(25,49)(26,47)(27,52)(29,56)(30,57)(31,55)(36,60)(37,61)(38,62)(39,63)(40,59)(41,64)(42,65)(44,69)(45,70)(46,68)(50,72)(51,71)(53,74)(54,73)(58,75)(66,76)(67,77);(17,32)(23,47)(28,55)(33,35)(34,59)(43,68)(48,64)(49,71)(52,73)(56,65)(57,75)(60,63)(61,76)(62,77)(69,70)(72,74);(14,18,21,20)(15,24,26,41)(16,29,31,42)(17,33,32,35)(19,36,40,39)(22,44,46,45)(23,48,47,64)(25,37,51,66)(27,53,54,50)(28,56,55,65)(30,38,58,67)(34,60,59,63)(43,69,68,70)(49,61,71,76)(52,74,73,72)(57,62,75,77)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55623);