Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2148 |
| 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 | |
220 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 220 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)(38,39,41,45,43,48,55,47)(40,49,52,63,53,66,50,65)(42,56,54,46,58,62,67,64)(44,57,60,51,61,68,59,69);(38,40)(39,44)(41,50)(42,57)(43,53)(45,59)(46,63)(47,60)(48,61)(49,62)(51,67)(52,55)(54,69)(56,66)(58,68)(64,65);(38,41,43,55)(39,45,48,47)(40,50,53,52)(42,54,58,67)(44,59,61,60)(46,62,64,56)(49,65,66,63)(51,57,69,68)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12646, C37×#12651, C37×#12740, … |
| maximal normal subgroup of | C37×#12646, C37×#12651, C37×#12740, … |
| maximal quotient of | C37×#11597, C37×#11598, C37×#11651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 880);