Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#2082 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C23×C29 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C4×C29 ◁ G | |
184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 928 |
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)(30,31,35,40,36,41,34,39)(32,42,45,37,46,53,44,54)(33,47,49,55,50,56,51,38)(43,58,59,61,48,52,60,57);(30,32,36,46)(31,37,41,54)(33,48,50,43)(34,45,35,44)(38,52,55,58)(39,53,40,42)(47,61,56,57)(49,59,51,60);(30,33,34,49,36,50,35,51)(31,38,39,47,41,55,40,56)(32,43,44,59,46,48,45,60)(37,52,42,57,54,58,53,61)667 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#2082, C29×#23652, C29×#24987, … |
| maximal normal subgroup of | C2×C29×#2082, C29×#23652, C29×#24987, … |
| maximal quotient of | C29×#10632, C29×#10633, C29×#10635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 814);