Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C23×C4 |
| direct product | C29×#3135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C29 |
| socle | C22×C29 | C22×D4 |
| prime radical | C29×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C2×C42×C29 ◁ G | |
392 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 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)(32,35)(33,34)(36,37)(38,39)(40,49)(41,45)(42,47)(43,52)(44,48)(46,51)(50,53);(31,34)(35,37)(38,40,43,41,44,51,42,50)(39,45,47,49,48,53,52,46);(30,32,33,36)(31,35,34,37)(38,41,43,51,44,50,42,40)(39,46,47,45,48,49,52,53);(38,42,44,43)(39,47,48,52)(40,50,51,41)(45,49,53,46)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#9005, C29×#9008, C29×#10033, … |
| maximal normal subgroup of | C29×#9005, C29×#9008, C29×#10033, … |
| maximal quotient of | C29×#20762, C29×#21086, C29×#21199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1867);