Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#12716 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #1781 |
| commutator | C22×C4 | C22×C4×C13 |
| Frattini, cosocle | C2×#41 | C23×C13 |
| prime radical | C13×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C13×H(5,𝔽2) ◁ G | |
976 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)(16,26)(17,30,18,23)(19,24)(20,25)(21,27,37,32)(22,31,39,33)(28,41)(29,40)(34,38)(35,44)(36,42,45,43);(14,16)(15,21)(17,27)(18,32)(19,28)(20,29)(22,35)(23,40)(24,36)(25,37)(26,30)(31,42)(33,43)(34,44)(38,45)(39,41);(14,17,20,18)(15,22,25,39)(16,27,29,32)(19,33,34,31)(21,35,37,41)(23,38,30,24)(26,36,40,45)(28,43,44,42)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5704);