Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#504 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C53 | C22 |
| commutator | C2 | C22×C8×C53 |
| Frattini, cosocle | C8 | C23×C53 |
| socle | C2×C53 | C22×C8 |
| prime radical | #41×C53 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×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)(54,55,57,61,58,62,67,73,59,63,68,74,69,75,79,83)(56,60,65,71,66,72,78,82,64,70,76,80,77,81,84,85);(54,56)(55,60)(57,65)(58,66)(59,64)(61,71)(62,72)(63,70)(67,78)(68,76)(69,77)(73,82)(74,80)(75,81)(79,84)(83,85);(56,64)(60,70)(65,76)(66,77)(71,80)(72,81)(78,84)(82,85)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 185);