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×#1930 |
| 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 | |
212 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 49 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)(30,31,33,32)(35,39)(36,42)(37,44)(40,47)(41,46)(45,48);(30,32,33,31)(34,35,37,39,38,41,44,46)(36,40,43,45,42,48,49,47);(34,36,37,43,38,42,44,49)(35,40,39,45,41,48,46,47)928 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1930, C29×#21075, C29×#21076, … |
| maximal normal subgroup of | C2×C29×#1930, C29×#21075, C29×#21076, … |
| maximal quotient of | C29×#9549, C29×#9551, C29×#9561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 662);