Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#12670 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | D8 |
| commutator | C8 | C2×C42×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C4×D8 |
| prime radical | C23×C13 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C22×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,23,29,24,30,22,28,25,31,39,47,38,46,21,27)(20,32,35,26,36,43,40,44,37,48,33,45,41,42,34,49);(14,15,16,17)(18,20)(19,26)(21,33)(22,35)(23,40)(24,41)(25,37)(27,42)(28,43)(29,48)(30,49)(31,45)(32,47)(34,39)(36,38)(44,46);(18,21,38,39,25,22,24,23)(19,27,46,47,31,28,30,29)(20,33,36,34,37,35,41,40)(26,42,44,32,45,43,49,48)1144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5658);