Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C2×C4 |
| direct product | C37×#1929 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C2×D4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C22×C37 | C4×D4 |
| prime radical | C2×#41×C37 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ C22×C8×C37 ◁ G | |
224 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 96 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)(39,40)(42,43,46,50,47,45,54,49)(44,48,52,56,53,51,57,55);(38,39,41,40)(42,44,46,52,47,53,54,57)(43,48,50,56,45,51,49,55);(38,40,41,39)(42,45)(43,47)(44,51)(46,49)(48,53)(50,54)(52,55)(56,57)1406 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#1929, C37×#21070, C37×#21071, … |
| maximal normal subgroup of | C2×C37×#1929, C37×#21070, C37×#21071, … |
| maximal quotient of | C37×#9541, C37×#9542, C37×#9543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 661);