Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42×C13 · Schur multiplier (H2) = C28×C4 |
| direct product | C13×#60058 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | C24 |
| commutator | C2 | C23×C42×C13 |
| Frattini, cosocle | C2×C4 | C25×C13 |
| socle | C22×C13 | C24×C4 |
| prime radical | C22×C13×#41 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ G | |
3178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 5 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,22,26,23,27,33,39)(20,24,30,36,29,35,42,46)(21,25,32,38,31,37,43,47)(28,34,41,45,40,44,48,49);(14,15,16,17)(18,20)(19,24)(21,28)(22,30)(23,29)(25,34)(26,36)(27,35)(31,40)(32,41)(33,42)(37,44)(38,45)(39,46)(43,48)(47,49);(18,21)(19,25)(20,28)(22,32)(23,31)(24,34)(26,38)(27,37)(29,40)(30,41)(33,43)(35,44)(36,45)(39,47)(42,48)(46,49);(21,31)(25,37)(28,40)(32,43)(34,44)(38,47)(41,48)(45,49);(20,29)(24,35)(28,40)(30,42)(34,44)(36,46)(41,48)(45,49)1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 53046);