Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C25 |
| direct product | C29×#1791 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | #120 |
| commutator | C23 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C2×C29 | #379 |
| prime radical | C29×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C22×C4×C29 ◁ G | |
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 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)(30,31,34,39,35,38,47,52)(32,40,43,53,42,36,46,49)(33,37,45,50,44,51,57,61)(41,48,56,59,55,54,58,60);(30,32)(31,36)(33,41)(34,46)(35,42)(37,48)(38,40)(39,53)(43,47)(44,55)(45,58)(49,52)(50,60)(51,54)(56,57)(59,61);(30,33)(31,37)(32,41)(34,45)(35,44)(36,48)(38,51)(39,50)(40,54)(42,55)(43,56)(46,58)(47,57)(49,59)(52,61)(53,60)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#13045, C29×#13046, C29×#13051, … |
| maximal normal subgroup of | C29×#13045, C29×#13046, C29×#13051, … |
| maximal quotient of | C29×#8316, C29×#8317, C29×#8348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 523);