Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#1905 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #386 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C2×#41×C29 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C4×C29 ◁ G | |
310 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 108 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)(30,31,35,39,36,41,34,40)(32,42,45,52,46,55,44,37)(33,47,49,38,50,53,48,54)(43,51,58,60,59,61,57,56);(30,32)(31,37,41,52)(33,43)(34,44)(35,45)(36,46)(38,51,54,61)(39,42,40,55)(47,56,53,60)(48,57)(49,58)(50,59);(30,33,36,50)(31,38,41,54)(32,43,46,59)(34,49,35,48)(37,51,52,61)(39,47,40,53)(42,56,55,60)(44,58,45,57)841 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1905, C29×#20998, C29×#22284, … |
| maximal normal subgroup of | C2×C29×#1905, C29×#20998, C29×#22284, … |
| maximal quotient of | C29×#9088, C29×#9090, C29×#9092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 637);