Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C37 · Schur multiplier (H2) = C4 |
| direct product | C37×#2164 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C37 | C23 |
| commutator | C22 | C22×C8×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle | C22×C37 | C22×C8 |
| prime radical | C23×C37 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C37 ◁ G | |
138 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 2368 |
A minimal faithful permutation representation — 3 generators on 77 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)(40,43)(41,42)(44,45)(46,47,49,53,50,54,59,65,51,55,60,66,61,67,71,75)(48,52,57,63,58,64,70,74,56,62,68,72,69,73,76,77);(39,42)(43,45)(46,48)(47,52)(49,57)(50,58)(51,56)(53,63)(54,64)(55,62)(59,70)(60,68)(61,69)(65,74)(66,72)(67,73)(71,76)(75,77);(38,40,41,44)(39,43,42,45)(48,56)(52,62)(57,68)(58,69)(63,72)(64,73)(70,76)(74,77)2072 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#2164, C37×#33419, C37×#33420, … |
| maximal normal subgroup of | C2×C37×#2164, C37×#33419, C37×#33420, … |
| maximal quotient of | C37×#12592, C37×#12600, C37×#12654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 896);