Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C13 · Schur multiplier (H2) = C2×C8 |
| direct product | C13×#8268 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C13 | C2×D4 |
| commutator | C4 | C2×C4×C8×C13 |
| Frattini, cosocle | C2×C42 | C23×C13 |
| socle | C22×C13 | C4×C4⋊C4 |
| prime radical | C23×C13 | #117 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(23,27,29,32)(24,30)(28,35,36,34)(31,37);(14,15,16,18,17,19,20,21)(22,23)(24,28)(25,32)(26,29)(27,33)(30,36)(31,35)(34,37);(22,24,25,31,26,30,33,37)(23,28,32,35,29,36,27,34)1144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 1256);