Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#2647 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C29 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ G | |
360 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 200 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)(31,32)(35,36)(38,39)(40,43)(41,44)(42,45)(46,47,50,48)(49,52,53,51);(30,31,34,32)(33,35,37,36)(38,40)(39,41)(42,44)(43,45)(46,48,50,47)(49,51,53,52);(30,32,34,31)(33,35,37,36)(38,40)(39,43)(41,45)(42,44)(46,49,50,53)(47,51,48,52);(30,33,34,37)(31,35,32,36)(38,41,42,43)(39,44,45,40)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#29133, C29×#29134, C29×#29135, … |
| maximal normal subgroup of | C29×#29133, C29×#29134, C29×#29135, … |
| maximal quotient of | C29×#14266, C29×#14276, C29×#14277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1379);