Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#2402 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C29 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| socle, prime radical | C23×C29 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C29 ◁ G | |
400 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 3712 |
A minimal faithful permutation representation — 4 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)(32,35)(33,34)(36,37)(38,39)(40,43)(41,45)(42,44)(46,47,49,51)(48,50,52,53);(31,34)(35,37)(38,40,42,41)(39,43,44,45)(46,48)(47,50)(49,52)(51,53);(38,41,42,40)(39,43,44,45)(48,52)(50,53);(30,32,33,36)(31,35,34,37)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#10372, C29×#10373, C29×#21676, … |
| maximal normal subgroup of | C29×#10372, C29×#10373, C29×#21676, … |
| maximal quotient of | C29×#13846, C29×#13848, C29×#13854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1134);