Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C22×C4 |
| direct product | C29×#1979 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #121 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C23×C29 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ G | |
276 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 65 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)(31,33)(34,35,39,44,40,43,38,52)(36,41,47,56,46,57,53,55)(37,48,50,58,51,62,49,42)(45,59,61,54,60,64,63,65);(30,31,32,33)(34,36,38,47,40,46,39,53)(35,41,44,55,43,57,52,56)(37,45,49,61,51,60,50,63)(42,54,48,59,58,65,62,64);(34,37,39,50,40,51,38,49)(35,42,52,62,43,58,44,48)(36,45,53,63,46,60,47,61)(41,54,56,64,57,65,55,59)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#13019, C29×#13020, C29×#13023, … |
| maximal normal subgroup of | C29×#13019, C29×#13020, C29×#13023, … |
| maximal quotient of | C29×#9669, C29×#9671, C29×#9725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 711);