Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C37 | C2×#99 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C2×Q8 | C23×C37 |
| prime radical | C2×#41×C37 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C37 ◁ C2×C4×C37 ◁ C2×#41×C37 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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)(38,39,43,48,44,49,41,46)(40,50,53,68,54,65,51,67)(42,56,55,66,58,47,59,64)(45,60,61,57,62,69,63,52);(38,40,44,54)(39,45,49,62)(41,53,43,51)(42,57,58,52)(46,63,48,61)(47,65,56,50)(55,60,59,69)(64,67,66,68);(38,41,44,43)(39,46,49,48)(40,51,54,53)(42,55,58,59)(45,61,62,63)(47,64,56,66)(50,67,65,68)(52,60,57,69)851 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#12712, C37×#12713, C37×#12721, … |
| maximal normal subgroup of | C37×#12712, C37×#12713, C37×#12721, … |
| maximal quotient of | C37×#11434, C37×#11450, C37×#11481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 867);