Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C23×C4 |
| direct product | C29×#2804 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| prime radical | C22×D4×C29 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ G | |
480 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 1856 |
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)(31,34)(32,36)(38,39,41,42)(40,43,44,45)(46,47)(48,51)(49,50)(52,53);(31,34)(35,37)(39,42)(40,44)(47,49)(48,52);(30,31)(32,35)(33,34)(36,37)(38,39)(40,43)(41,42)(44,45)(46,48,50,52)(47,51,49,53);(30,32,33,36)(31,35,34,37)(38,40)(39,43)(41,44)(42,45)(46,49,50,47)(48,53,52,51)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#7745, C29×#7755, C29×#7782, … |
| maximal normal subgroup of | C29×#7745, C29×#7755, C29×#7782, … |
| maximal quotient of | C29×#14440, C29×#14457, C29×#14471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1536);