Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#508 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | D8 |
| commutator, Frattini, cosocle | C8 | C23×C53 |
| socle | C2×C53 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C2×C8×C53 ◁ G | |
84 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)(56,65)(57,62)(58,70)(59,74)(60,64)(61,67)(63,72)(66,81)(68,82)(69,80)(71,76)(73,84)(75,79)(77,85)(78,83);(54,56)(55,61)(57,66)(58,71)(59,75)(60,68)(62,76)(63,79)(64,78)(65,70)(67,72)(69,83)(73,85)(74,77)(80,81)(82,84);(54,57,60,59)(55,62,64,74)(56,66,68,75)(58,69,73,72)(61,76,78,77)(63,70,80,84)(65,81,82,79)(67,71,83,85)1166 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 189);