Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C53 |
| prime radical | C2×D4×C53 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ G | |
112 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 73 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)(55,57)(58,59)(60,66)(61,63)(62,70)(64,68)(65,69)(67,73)(71,72);(54,55,56,57)(59,63)(64,72)(65,70)(66,73);(58,60,61,67)(59,64,63,72)(62,71,69,68)(65,73,70,66)1007 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 145);