Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C22×C4 |
| direct product | C37×#2140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C2×D8 |
| commutator | C8 | C22×C4×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| prime radical | C22×D8×C37 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C22×C4×C37 ◁ C22×C8×C37 ◁ G | |
260 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 3 generators on 57 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)(44,49)(45,48)(46,55)(47,51)(50,53)(52,56)(54,57);(43,48)(44,46)(45,53)(49,50)(51,57)(52,54)(55,56);(42,44,47,46)(43,49,51,55)(45,52,54,53)(48,56,57,50)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12647, C37×#12741, C37×#12750, … |
| maximal normal subgroup of | C37×#12647, C37×#12741, C37×#12750, … |
| maximal quotient of | C37×#11593, C37×#11649, C37×#11661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 872);