Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#12713 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #1781 |
| commutator | C22×C4 | C22×C4×C13 |
| Frattini, cosocle | C2×#41 | C23×C13 |
| prime radical | C13×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C13×H(5,𝔽2) ◁ G | |
986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,25)(17,29,31,22)(18,23)(19,24)(20,26,36,41)(21,30,37,32)(27,34)(28,39)(33,38)(35,40,45,42)(43,44);(14,16,19,28)(15,20,24,36)(17,26,31,41)(18,27,33,43)(21,34,37,44)(22,25,29,39)(23,35,38,45)(30,40,32,42);(14,17)(15,21)(16,26)(18,32)(19,31)(20,34)(22,23)(24,37)(25,35)(27,42)(28,41)(29,38)(30,33)(36,44)(39,45)(40,43)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5701);