Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13×C16 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13549 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C13×C32 | C22 |
| commutator | C2 | C2×C4×C13×C16 |
| Frattini, cosocle | C2×C16 | C23×C13 |
| socle | C22×C13 | C22×C16 |
| prime radical | C23×C13 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13×C32 ◁ G | |
238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,21,26,22,27,34,44,23,28,35,45,37,47,56,66,24,29,36,46,38,48,57,67,39,49,58,68,59,69,74,79)(20,25,31,41,32,42,53,63,33,43,54,64,55,65,73,78,30,40,50,60,51,61,70,75,52,62,71,76,72,77,80,81);(14,15,16,17)(18,20)(19,25)(21,31)(22,32)(23,33)(24,30)(26,41)(27,42)(28,43)(29,40)(34,53)(35,54)(36,50)(37,55)(38,51)(39,52)(44,63)(45,64)(46,60)(47,65)(48,61)(49,62)(56,73)(57,70)(58,71)(59,72)(66,78)(67,75)(68,76)(69,77)(74,80)(79,81);(20,30)(25,40)(31,50)(32,51)(33,52)(41,60)(42,61)(43,62)(53,70)(54,71)(55,72)(63,75)(64,76)(65,77)(73,80)(78,81)2080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6537);