Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #120 |
| commutator, Frattini, cosocle | C23 | C23×C53 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C23×C53 ◁ G | |
198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 424 |
A minimal faithful permutation representation — 3 generators on 61 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)(58,61);(54,55)(56,58)(57,59)(60,61);(54,56)(55,58)(57,60)(59,61)848 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 138);