Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#12781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #1790 |
| commutator | C2×Q8 | C22×C4×C13 |
| Frattini, cosocle | C22×Q8 | C23×C13 |
| prime radical | C13×#3545 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C23×C13 ◁ C22×Q8×C13 ◁ G | |
758 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)(14,15,19,22,20,25,18,24)(16,26,28,44,29,40,34,43)(17,23,32,41,33,42,31,39)(21,35,37,27,38,45,36,30);(14,16)(15,21)(17,30)(18,34)(19,28)(20,29)(22,36)(23,40)(24,37)(25,38)(26,42)(27,33)(31,45)(32,35)(39,43)(41,44);(15,22)(18,19)(21,36)(23,41)(24,25)(26,43)(28,34)(31,32)(35,45)(37,38)(39,42)(40,44)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5769);