Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#1888 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #120 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C22×D4×C37 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C22×C4×C37 ◁ G | |
344 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 3 generators on 57 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)(42,43,48,51,47,44,46,50)(45,49,55,57,54,52,53,56);(38,39,40,41)(42,44,47,43)(45,52,54,49)(46,50,48,51)(53,56,55,57);(42,45,47,54)(43,49,44,52)(46,55,48,53)(50,57,51,56)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7481, C37×#7489, C37×#12774, … |
| maximal normal subgroup of | C37×#7481, C37×#7489, C37×#12774, … |
| maximal quotient of | C37×#8960, C37×#8964, C37×#9002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 620);