Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#575 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C53 |
| socle | C2×C53 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ G | |
262 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 4 generators on 69 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)(56,63)(57,62)(58,66)(59,61)(60,64)(65,69)(67,68);(54,56)(55,60)(57,64)(58,67)(59,65)(61,68)(62,69)(63,66);(55,61)(60,68)(62,66)(63,69);(54,57,59,58)(55,62,61,66)(56,64,65,67)(60,69,68,63)1166 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 256);