Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#9853 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | #127 |
| commutator | C42 | C22×C4×C13 |
| Frattini, cosocle | C2×C42 | C23×C13 |
| socle | C22×C13 | #390 |
| prime radical | C23×C13 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,21,19,22,17,27)(16,23,26,28,24,29,25,20)(30,31)(32,38)(33,35)(34,42)(36,40)(37,41)(39,44)(43,45);(14,16)(15,20,22,28)(17,25)(18,26)(19,24)(21,23,27,29)(31,35)(32,39)(37,42)(40,45);(16,24)(20,28)(23,29)(25,26)(30,32,34,40,33,39,41,45)(31,36,37,38,35,43,42,44)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 2841);