Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C37 · Schur multiplier (H2) = C2 |
| direct product | C37×#1353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #99 |
| commutator | C22×C4 | C2×C4×C37 |
| Frattini, cosocle | C2×C4⋊C4 | C22×C37 |
| prime radical | C23×C37 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C23×C37 ◁ C2×C4⋊C4×C37 ◁ G | |
116 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 40 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 — 2 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)(39,42,43,44)(46,47,49,53,50,52,56,59)(48,51,55,57,54,58,61,60);(38,39)(40,44)(41,43)(42,45)(46,48)(47,51,52,58)(49,55)(50,54)(53,60,59,57)(56,61)851 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C37×#1353, C37×#11721, C37×#11722, … |
| maximal normal subgroup of | C2×C37×#1353, C37×#11721, C37×#11722, … |
| maximal quotient of | C37×#7122, C37×#7132, C37×#7251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 85);