Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13173 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #466 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1942 |
| prime radical | C23×C13 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
262 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)(14,15,18,26,21,27,20,24)(16,28,32,45,33,50,30,22)(17,34,36,66,39,67,38,65)(19,41,40,52,42,51,35,25)(23,46,48,69,49,68,47,37)(29,55,59,74,60,75,57,64)(31,61,62,73,63,44,56,54)(43,58,53,71,70,77,72,76);(14,16,18,32,21,33,20,30)(15,22,26,28,27,45,24,50)(17,29,36,59,39,60,38,57)(19,31,40,62,42,63,35,56)(23,43,48,53,49,70,47,72)(25,44,41,54,52,61,51,73)(34,64,66,55,67,74,65,75)(37,58,46,71,69,77,68,76);(14,17,35,68,20,38,42,69,21,39,40,46,18,36,19,37)(15,23,41,66,26,48,52,67,27,49,51,65,24,47,25,34)(16,29,56,76,30,57,63,77,33,60,62,71,32,59,31,58)(22,43,54,55,28,53,61,74,45,70,73,75,50,72,44,64)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6161);