Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C37 | C2×#99 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C2×Q8 | C23×C37 |
| prime radical | C2×#41×C37 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C37 ◁ C23×C37 ◁ C22×Q8×C37 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 1184 |
A minimal faithful permutation representation — 3 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)(38,39,42,46,43,47,58,65)(40,48,51,66,49,62,56,68)(41,52,54,64,55,45,57,63)(44,59,61,53,60,67,69,50);(38,40)(39,44)(41,53)(42,56)(43,49)(45,62)(46,61)(47,60)(48,52)(50,55)(51,58)(54,59)(57,67)(63,68)(64,66)(65,69);(40,49)(44,60)(48,62)(50,53)(51,56)(59,67)(61,69)(66,68)851 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#12709, C37×#12710, C37×#12729, … |
| maximal normal subgroup of | C37×#12709, C37×#12710, C37×#12729, … |
| maximal quotient of | C37×#11432, C37×#11448, C37×#11480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 865);