Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C2×C4 |
| direct product | C29×#1496 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C42 | C23×C29 |
| socle | C22×C29 | C2×C22⋊C4 |
| prime radical | C29×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C2×C42×C29 ◁ G | |
246 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 53 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)(32,36)(33,35)(34,37)(38,39,41,45,42,44,48,51)(40,43,47,49,46,50,52,53);(31,35)(32,33)(36,37)(38,40)(39,43)(41,47)(42,46)(44,50)(45,49)(48,52)(51,53);(30,32,34,33)(31,36,37,35)(39,44)(40,46)(45,51)(47,52)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#8489, C29×#8500, C29×#8503, … |
| maximal normal subgroup of | C29×#8489, C29×#8500, C29×#8503, … |
| maximal quotient of | C29×#7698, C29×#7699, C29×#7700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 228);