Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C53 |
| prime radical | C23×C53 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ G | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 77 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)(56,60)(58,61)(62,63,66,69)(64,67,72,70)(65,73,74,68)(71,76,77,75);(54,55,57,59)(56,58,60,61)(62,63)(64,70)(65,68)(66,69)(67,72)(71,76)(73,74)(75,77);(54,56)(55,58)(57,60)(59,61)(62,64)(63,67)(65,71)(66,72)(68,75)(69,70)(73,76)(74,77);(62,65,66,74)(63,68,69,73)(64,71,72,77)(67,75,70,76)1166 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 222);