Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8×C13 · Schur multiplier (H2) = C29 |
| direct product | C13×#62624 |
| subgroup | G/H | |
|---|---|---|
| center | C13×C16 | C24 |
| commutator | C2 | C24×C8×C13 |
| Frattini, cosocle | C8 | C25×C13 |
| socle | C2×C13 | C24×C8 |
| prime radical | C13×#585 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C13×C16 ◁ 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)(14,15,18,23,19,24,33,43,20,25,34,44,35,45,55,65)(16,21,28,38,29,39,51,61,27,37,49,59,50,60,69,74)(17,22,31,41,32,42,54,64,30,40,52,62,53,63,70,75)(26,36,47,57,48,58,68,73,46,56,66,71,67,72,76,77);(14,16)(15,21)(17,26)(18,28)(19,29)(20,27)(22,36)(23,38)(24,39)(25,37)(30,46)(31,47)(32,48)(33,51)(34,49)(35,50)(40,56)(41,57)(42,58)(43,61)(44,59)(45,60)(52,66)(53,67)(54,68)(55,69)(62,71)(63,72)(64,73)(65,74)(70,76)(75,77);(14,17)(15,22)(16,26)(18,31)(19,32)(20,30)(21,36)(23,41)(24,42)(25,40)(27,46)(28,47)(29,48)(33,54)(34,52)(35,53)(37,56)(38,57)(39,58)(43,64)(44,62)(45,63)(49,66)(50,67)(51,68)(55,70)(59,71)(60,72)(61,73)(65,75)(69,76)(74,77);(17,30)(22,40)(26,46)(31,52)(32,53)(36,56)(41,62)(42,63)(47,66)(48,67)(54,70)(57,71)(58,72)(64,75)(68,76)(73,77);(16,27)(21,37)(26,46)(28,49)(29,50)(36,56)(38,59)(39,60)(47,66)(48,67)(51,69)(57,71)(58,72)(61,74)(68,76)(73,77)1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55612);