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×#1932 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #121 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C2×C37 | #388 |
| prime radical | C2×#41×C37 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ G | |
216 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 592 |
A minimal faithful permutation representation — 3 generators on 53 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)(38,39,42,46,43,47,41,44)(40,48,51,52,50,45,49,53);(39,44,47,46)(45,52,48,53);(38,40,41,49,43,50,42,51)(39,45,46,53,47,48,44,52)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#1932, C37×#21080, C37×#21558, … |
| maximal normal subgroup of | C2×C37×#1932, C37×#21080, C37×#21558, … |
| maximal quotient of | C37×#9549, C37×#9550, C37×#9551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 664);