Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C29 |
| direct product | C53×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | C24 |
| commutator, Frattini, cosocle | C2 | C25×C53 |
| socle | C2×C53 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ G | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 5 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)(56,62)(57,61)(58,60)(59,63)(64,68)(65,67)(66,69);(54,56)(55,59)(57,64)(58,63)(60,62)(61,66)(65,69)(67,68);(56,63)(59,62)(64,69)(66,68);(55,60)(59,62)(61,67)(66,68);(54,57,58,65)(55,61,60,67)(56,64,63,69)(59,66,62,68)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 266);