Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#1881 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | #120 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C2×C29 | #386 |
| prime radical | C29×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×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)(32,41,44,52)(33,45)(34,39)(35,49)(36,40)(37,43,53,48)(38,46)(42,57,60,51)(47,54)(50,55)(56,58,61,59);(30,32,34,48,36,44,35,43)(31,37,39,52,40,53,49,41)(33,42,46,59,47,60,50,58)(38,51,54,61,55,57,45,56);(30,33)(31,38)(32,42)(34,46)(35,50)(36,47)(37,51)(39,54)(40,55)(41,56)(43,58)(44,60)(45,49)(48,59)(52,61)(53,57)928 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1881, C29×#20939, C29×#21477, … |
| maximal normal subgroup of | C2×C29×#1881, C29×#20939, C29×#21477, … |
| maximal quotient of | C29×#8953, C29×#8954, C29×#8955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 613);