Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#1915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C37 | #386 |
| commutator | C23 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C2×#41×C37 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C37 ◁ C22×C4×C37 ◁ G | |
308 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 308 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)(38,39,42,47,43,46,48,61)(40,44,51,58,50,59,55,62)(41,52,54,63,53,45,56,60)(49,57,65,68,64,67,66,69);(38,40,48,51,43,50,42,55)(39,44,47,62,46,59,61,58)(41,49,54,66,53,64,56,65)(45,57,63,68,52,67,60,69);(38,41)(39,45)(40,49)(42,56)(43,53)(44,57)(46,52)(47,63)(48,54)(50,64)(51,66)(55,65)(58,69)(59,67)(60,61)(62,68)1073 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12757, C37×#12758, C37×#12770, … |
| maximal normal subgroup of | C37×#12757, C37×#12758, C37×#12770, … |
| maximal quotient of | C37×#9108, C37×#9110, C37×#9189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 647);