Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#1868 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #120 |
| commutator | C23 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C2×C37 | #385 |
| prime radical | C2×#41×C37 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ G | |
260 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 108 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)(38,39,42,46,43,47,53,59)(40,44,50,56,49,55,63,67)(41,45,52,58,51,57,64,68)(48,60,62,69,61,54,65,66);(38,40,42,50,43,49,53,63)(39,44,46,56,47,55,59,67)(41,48,52,62,51,61,64,65)(45,54,58,66,57,60,68,69);(38,41)(39,45)(40,48)(42,52)(43,51)(44,54)(46,58)(47,57)(49,61)(50,62)(53,64)(55,60)(56,66)(59,68)(63,65)(67,69)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12962, C37×#12963, C37×#12968, … |
| maximal normal subgroup of | C37×#12962, C37×#12963, C37×#12968, … |
| maximal quotient of | C37×#8851, C37×#8852, C37×#8861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 600);