Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C13×#3038 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C2×C4×Q8×C13 ◁ G | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,18)(19,20)(22,23,27,31)(24,28,26,30)(25,33,34,36)(29,35,37,32);(14,15,17,18)(16,20,21,19)(22,24,27,26)(23,28,31,30)(25,32,34,35)(29,36,37,33);(22,25)(23,29)(24,32)(26,35)(27,34)(28,36)(30,33)(31,37);(14,16,17,21)(15,19,18,20);(22,24,27,26)(23,30,31,28)(25,32,34,35)(29,33,37,36)715 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54312);