Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C37 · Schur multiplier (H2) = C25 |
| direct product | C37×#2912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C22×D4 |
| commutator | C4 | C23×C4×C37 |
| Frattini, cosocle | C2×C4 | C24×C37 |
| prime radical | C2×#41×C37 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C2×#41×C37 ◁ G | |
476 conjugacy classes of subgroups; only the 344 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 344 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) = 2368 |
A minimal faithful permutation representation — 4 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)(38,39,40,41)(43,47)(45,52)(48,53)(49,54)(51,56)(55,57);(42,43,45,47,46,49,52,54)(44,48,51,53,50,55,56,57);(42,44)(43,48)(45,51)(46,50)(47,53)(49,55)(52,56)(54,57);(44,50)(48,55)(51,56)(53,57)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12873, C37×#12874, C37×#12934, … |
| maximal normal subgroup of | C37×#12873, C37×#12874, C37×#12934, … |
| maximal quotient of | C37×#17788, C37×#17798, C37×#17811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 1644);