Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#12720 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #410 |
| commutator | C22×C4 | C22×C4×C13 |
| Frattini, cosocle | C2×#41 | C23×C13 |
| socle | C2×C13 | #1782 |
| prime radical | C13×#3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C42×C13 ◁ C4×C13×#41 ◁ G | |
694 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)(17,30,32,22)(18,23)(19,24)(20,27,36,26)(21,31,38,45)(28,37)(29,41)(33,39)(34,44)(35,43,40,42);(14,16)(15,20)(17,27)(18,28)(19,29)(21,37)(22,25)(23,35)(24,36)(26,32)(30,41)(31,43)(33,44)(34,38)(39,40)(42,45);(14,17)(15,21,24,38)(16,26)(18,31)(19,32)(20,34,36,37)(22,23,30,39)(25,40,41,35)(27,29)(28,42)(33,45)(43,44)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5708);