Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C24×C4 |
| direct product | C13×#17554 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C13 |
| Frattini, cosocle | C22×C4 | C24×C13 |
| socle | C22×C13 | C22×C22⋊C4 |
| prime radical | C2×C13×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C22×C42×C13 ◁ G | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,22,24,19)(17,20)(18,21)(23,28,29,26)(25,27)(30,31)(32,38,40,45)(33,37)(34,36)(35,39,43,41)(42,44);(14,16)(15,19)(17,23)(18,24)(20,26)(21,22)(25,29)(27,28)(30,32)(31,35)(33,41)(34,40)(36,43)(37,45)(38,44)(39,42);(31,36)(35,43)(37,44)(38,45);(14,17,18,25)(15,20,21,27)(16,23,24,29)(19,26,22,28)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 10542);