Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#525 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C53 |
| socle | C22×C53 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
330 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 4 generators on 65 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)(56,57)(58,59)(60,63)(61,62)(64,65);(59,62)(63,65);(55,57);(58,60,61,64)(59,63,62,65)1484 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 206);