Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#2178 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C37 | D8 |
| commutator | C8 | C22×C4×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle | C2×C37 | C4×D8 |
| prime radical | C37×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C37 ◁ C2×C8×C37 ◁ C4×C8×C37 ◁ G | |
146 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 76 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)(38,39,43,47,44,49,59,48)(40,50,52,61,54,67,53,69)(41,46,55,64,56,65,42,57)(45,60,62,68,63,58,66,51);(38,40)(39,45)(41,51)(42,58)(43,52)(44,54)(46,61)(47,62)(48,66)(49,63)(50,57)(53,59)(55,60)(56,68)(64,67)(65,69);(38,41,43,55,44,56,59,42)(39,46,47,64,49,65,48,57)(40,51,52,60,54,68,53,58)(45,61,62,67,63,69,66,50)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7387, C37×#7388, C37×#12046, … |
| maximal normal subgroup of | C37×#7387, C37×#7388, C37×#12046, … |
| maximal quotient of | C8×D16×C37, C8×SD32×C37, C8×Q32×C37, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 910);