Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2 |
| direct product | C53×#446 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C23 |
| commutator | C22 | C22×C4×C53 |
| Frattini, cosocle | C2×C4 | C23×C53 |
| socle, prime radical | C22×C53 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55)(56,61)(57,59)(58,60)(62,63,65,64,66,67,69,68);(54,56,58,57)(55,59,60,61);(54,57,58,56)(55,59,60,61)(62,64,65,67,66,68,69,63)1484 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 127);