Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2145 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #102 |
| commutator | C8 | C22×C4×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle | C2×C37 | C2×#102 |
| prime radical | C2×#135×C37 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ C22×C8×C37 ◁ G | |
224 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 100 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)(40,49,52,66)(41,45)(42,56)(43,48)(44,51,60,68)(46,54)(47,55)(50,65,53,62)(57,64,69,59)(58,61)(63,67);(38,40)(39,44)(41,50)(42,57)(43,52)(45,59)(46,62)(47,64)(48,60)(49,61)(51,67)(53,55)(54,68)(56,66)(58,69)(63,65);(38,41,43,55)(39,45,48,47)(40,50,52,53)(42,54,58,67)(44,59,60,64)(46,61,63,56)(49,65,66,62)(51,57,68,69)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12646, C37×#12649, C37×#12741, … |
| maximal normal subgroup of | C37×#12646, C37×#12649, C37×#12741, … |
| maximal quotient of | C37×#11597, C37×#11599, C37×#11600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 877);