Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#13100 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #1898 |
| commutator | C22×C4 | C22×C4×C13 |
| Frattini, cosocle | C2×#41 | C23×C13 |
| prime radical | C2×C13×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C13×H(5,𝔽2) ◁ G | |
672 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)(16,26,30,42,31,44,29,43)(17,32,27,40,34,23,35,37)(21,36,38,33,39,45,41,28);(14,16,27,36)(15,21,37,42)(17,33,19,29)(18,30,34,28)(20,31,35,45)(22,38,23,26)(24,41,32,44)(25,39,40,43);(15,22)(17,34)(18,19)(21,38)(23,37)(24,25)(26,42)(28,33)(29,30)(32,40)(39,41)(43,44)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6088);