Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C4 |
| direct product | C37×#1922 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C2×D4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C22×C37 | C4×D4 |
| prime radical | C23×C37 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ C22×C8×C37 ◁ G | |
186 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 186 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) = 61 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 3 generators on 61 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)(46,47,50,54,51,49,58,53)(48,52,56,60,57,55,61,59);(39,42)(43,45)(46,48,50,56,51,57,58,61)(47,52,54,60,49,55,53,59);(38,40,41,44)(39,43,42,45)(46,49)(47,51)(48,55)(50,53)(52,57)(54,58)(56,59)(60,61)1406 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#1922, C37×#21041, C37×#21042, … |
| maximal normal subgroup of | C2×C37×#1922, C37×#21041, C37×#21042, … |
| maximal quotient of | C37×#9533, C37×#9534, C37×#9535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 654);