Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C13 · Schur multiplier (H2) = C24×C4 |
| direct product | C13×#20510 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | C2×D4 |
| commutator | C4 | C22×C42×C13 |
| Frattini, cosocle | C22×C4 | C24×C13 |
| socle | C22×C13 | C2×C4×D4 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,23,28,24,26,22,36)(20,29,31,25,32,40,37,38)(21,27,34,42,35,41,33,46)(30,43,44,39,45,49,47,48);(14,15,17,16)(18,20)(19,25)(21,30)(22,37)(23,31)(24,32)(26,38)(27,39)(28,40)(29,36)(33,47)(34,44)(35,45)(41,48)(42,49)(43,46);(14,16,17,15)(19,26)(25,38)(27,41)(28,36)(29,40)(39,48)(42,46)(43,49);(18,21,23,34,24,35,22,33)(19,27,28,42,26,41,36,46)(20,30,31,44,32,45,37,47)(25,39,40,49,38,48,29,43)1144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 13498);