Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#1891 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #120 |
| commutator | C23 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C2×C37 | #386 |
| prime radical | C37×#525 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ G | |
342 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 108 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)(38,39,42,46,43,40,44,48)(41,47,50,53,49,45,51,52);(38,40,42,48,43,39,44,46)(41,47,51,52,49,45,50,53);(38,41)(39,45)(40,47)(42,51)(43,49)(44,50)(46,53)(48,52)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12755, C37×#12756, C37×#12757, … |
| maximal normal subgroup of | C37×#12755, C37×#12756, C37×#12757, … |
| maximal quotient of | C37×#8965, C37×#8966, C37×#8967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 623);