Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23×C53 |
| prime radical | #135×C53 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×C8×C53 ◁ G | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 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)(56,64)(57,68)(58,73)(59,61)(60,65)(62,71)(63,70)(66,84)(67,81)(69,79)(72,80)(74,78)(75,83)(76,85)(77,82);(54,56,59,67)(55,60,61,75)(57,69,72,85)(58,74,70,66)(62,78,77,84)(63,79,73,76)(64,80,81,68)(65,82,83,71);(55,61)(60,75)(62,77)(63,73)(64,81)(68,80)(76,79)(78,84)848 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 191);