Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#13243 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #466 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1970 |
| prime radical | C2×C13×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,28,33,52)(17,35)(18,25)(19,42)(20,44)(21,27)(22,31,47,40)(23,37)(24,29)(26,43)(30,45,60,71)(32,48,56,70)(34,46,62,55)(36,64)(38,49)(39,66)(41,53)(50,68)(51,57)(54,74,73,59)(58,75,67,63)(61,77,76,72)(65,69);(14,16,29,56,43,62,18,40,21,33,19,32,20,34,41,31)(15,22,25,46,44,70,24,52,27,47,53,55,26,48,42,28)(17,30,57,74,69,76,36,67,39,60,37,59,38,61,68,58)(23,45,35,63,64,77,49,73,51,71,66,75,50,72,65,54);(14,17,18,36,20,38,29,57,21,39,41,68,43,69,19,37)(15,23,42,65,26,50,53,66,27,51,24,49,44,64,25,35)(16,30,40,67,34,61,56,74,33,60,31,58,62,76,32,59)(22,45,28,54,48,72,55,75,47,71,52,73,70,77,46,63)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6231);