Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C2 |
| direct product | C29×#1704 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C29 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C29 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C42×C29 ◁ G | |
140 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 3712 |
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)(30,31,34,37)(32,38,40,45)(33,42,39,36)(35,43,44,41)(46,47)(48,54)(49,51)(50,58)(52,56)(53,57)(55,61)(59,60);(30,32,39,43,34,40,33,41)(31,35,42,45,37,44,36,38)(47,51)(52,60)(53,58)(54,61);(32,40)(35,44)(38,45)(41,43)(46,48,49,55)(47,52,51,60)(50,59,57,56)(53,61,58,54)667 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1704, C29×#19155, C29×#19158, … |
| maximal normal subgroup of | C2×C29×#1704, C29×#19155, C29×#19158, … |
| maximal quotient of | C29×#8657, C29×#9224, C29×#9228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 436);