Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C2×D8 |
| commutator | C8 | C22×C4×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| prime radical | C23×C37 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C22×C4×C37 ◁ C22×C8×C37 ◁ G | |
220 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 104 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)(44,53)(45,50)(46,59)(47,63)(48,52)(49,55)(51,61)(54,67)(56,73)(57,70)(58,68)(60,66)(62,69)(64,71)(65,72);(42,44,48,57)(43,49,52,65)(45,56,47,54)(46,60,62,71)(50,66,63,64)(51,67,68,73)(53,69,70,59)(55,58,72,61);(42,45,48,47)(43,50,52,63)(44,54,57,56)(46,58,62,61)(49,64,65,66)(51,59,68,69)(53,67,70,73)(55,71,72,60)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12648, C37×#12742, C37×#12751, … |
| maximal normal subgroup of | C37×#12648, C37×#12742, C37×#12751, … |
| maximal quotient of | C37×#11594, C37×#11650, C37×#11662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 873);