Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#7180 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | #106 |
| commutator | C8 | C4×C8×C13 |
| Frattini, cosocle | C2×C4×C8 | C22×C13 |
| socle | C2×C13 | #1279 |
| prime radical | C23×C13 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C22×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,24,20,26,36,41,21,27,35,23,19,25,17,34)(16,28,30,39,32,44,29,40,33,42,37,38,31,43,45,22);(14,16)(15,22,25,38,27,40,26,39)(17,29)(18,37)(19,31)(20,32)(21,33)(23,28,41,43,24,42,34,44)(30,35)(36,45)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 168);