Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C2×C4 |
| direct product | C37×#2251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C37 |
| prime radical | C2×D8×C37 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C42×C37 ◁ C4×C8×C37 ◁ G | |
178 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 2368 |
A minimal faithful permutation representation — 3 generators on 73 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)(39,41)(42,43,47,51)(44,52,54,66)(45,57,59,72)(46,60,58,50)(48,61,62,55)(49,64,65,53)(56,71,70,68)(63,69,67,73);(42,44,53,69,46,56,59,61,47,54,64,73,58,70,45,55)(43,48,57,68,50,63,65,66,51,62,72,71,60,67,49,52);(38,39,40,41)(44,54)(48,62)(52,66)(55,61)(56,70)(63,67)(68,71)(69,73)962 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#12326, C37×#12327, C37×#12338, … |
| maximal normal subgroup of | C37×#12326, C37×#12327, C37×#12338, … |
| maximal quotient of | C37×#12507, C37×#12508, C37×#12512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 983);