Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C2 |
| direct product | C29×#1592 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C42 | C23×C29 |
| socle | C22×C29 | C4×D4 |
| prime radical | C2×D4×C29 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C4×C8×C29 ◁ G | |
162 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 45 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)(32,34)(33,35)(36,37)(38,39,42,40,41,43,45,44);(31,34)(32,37)(35,36)(39,43)(40,44);(31,35)(34,36)(38,40,42,43,41,44,45,39)928 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1592, C29×#17890, C29×#17891, … |
| maximal normal subgroup of | C2×C29×#1592, C29×#17890, C29×#17891, … |
| maximal quotient of | C29×#8208, C29×#8209, C29×#8286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 324);