Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C2×C4 |
| direct product | C37×#1521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | #121×C37 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G | |
218 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 88 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)(39,44,46,58)(40,49)(41,54)(45,60,61,66)(47,59,48,55)(50,65,62,57)(52,68)(53,64)(56,67)(63,69);(38,39)(40,50)(41,55)(42,47)(43,48)(44,56)(45,53)(46,51)(49,61)(52,65)(54,58)(57,68)(59,67)(60,69)(62,64)(63,66);(38,40,51,64)(39,45,46,61)(41,52,67,68)(42,53,43,49)(44,57,58,65)(47,50,48,62)(54,69,56,63)(55,60,59,66)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#8500, C37×#8501, C37×#8502, … |
| maximal normal subgroup of | C37×#8500, C37×#8501, C37×#8502, … |
| maximal quotient of | C37×#7580, C37×#7581, C37×#7584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 253);