Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13×C16 · Schur multiplier (H2) = C25 |
| direct product | C13×#33496 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | C24 |
| commutator | C2 | C23×C13×C16 |
| Frattini, cosocle | C2×C8 | C24×C13 |
| socle | C22×C13 | C23×C8 |
| prime radical | C23×C13 | C22×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ G | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1664 |
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)(17,20)(18,21)(19,24)(23,28)(25,27)(26,29)(30,31,32,35,33,36,38,41,34,37,39,42,40,43,44,45);(14,16,18,24)(15,19,21,22)(17,25,26,23)(20,28,29,27);(14,17,18,26)(15,20,21,29)(16,23,24,25)(19,27,22,28);(15,21)(19,22)(20,29)(27,28)1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26484);