Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8×C13 · Schur multiplier (H2) = C24×C4 |
| direct product | C13×#33326 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | C24 |
| commutator | C2 | C22×C4×C8×C13 |
| Frattini, cosocle | C2×C8 | C24×C13 |
| socle | C22×C13 | C23×C8 |
| prime radical | C2×C13×#41 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ G | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,21,26,22,27,31,39,23,24,32,35,33,36,43,46)(20,25,29,37,30,38,42,47,28,34,40,44,41,45,48,49);(14,15,16,17)(19,24)(25,34)(26,35)(27,36)(37,44)(38,45)(39,46)(47,49);(18,20)(19,25)(21,29)(22,30)(23,28)(24,34)(26,37)(27,38)(31,42)(32,40)(33,41)(35,44)(36,45)(39,47)(43,48)(46,49);(20,28)(25,34)(29,40)(30,41)(37,44)(38,45)(42,48)(47,49)1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26314);