Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#1662 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C29 |
| prime radical | C29×#562 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C42×C29 ◁ G | |
196 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 61 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)(30,31,35,39)(32,40,42,45)(33,38,34,37)(36,43,44,41)(46,47,50,53)(48,54,56,61)(49,57,59,52)(51,58,60,55);(30,32)(31,36)(33,43)(34,41)(35,42)(37,45)(38,40)(39,44)(46,48,50,56)(47,51,53,60)(49,58,59,55)(52,61,57,54);(30,32)(31,36)(33,41)(34,43)(35,42)(37,40)(38,45)(39,44)(49,59)(52,57)(54,61)(55,58)580 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#7525, C2×C29×#1662, C29×#18987, … |
| maximal normal subgroup of | C29×#7525, C2×C29×#1662, C29×#18987, … |
| maximal quotient of | C29×#7762, C29×#7769, C29×#7773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 394);