Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2113 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C37 | C23 |
| commutator | C22 | C22×C8×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle | C22×C37 | C22×C8 |
| prime radical | C2×#41×C37 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C37 ◁ G | |
228 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 120 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)(38,39)(40,41)(42,43,45,49,46,50,55,61,47,51,56,62,57,63,67,71)(44,48,53,59,54,60,66,70,52,58,64,68,65,69,72,73);(39,41)(42,44)(43,48)(45,53)(46,54)(47,52)(49,59)(50,60)(51,58)(55,66)(56,64)(57,65)(61,70)(62,68)(63,69)(67,72)(71,73);(44,52)(48,58)(53,64)(54,65)(59,68)(60,69)(66,72)(70,73)2072 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7207, C37×#7208, C37×#7227, … |
| maximal normal subgroup of | C37×#7207, C37×#7208, C37×#7227, … |
| maximal quotient of | C37×#11419, C37×#11441, C37×#11461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 845);