Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C29 · Schur multiplier (H2) = C29 |
| direct product | C22×C29×#123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C29 | C23 |
| commutator, Frattini, cosocle | C22 | C25×C29 |
| prime radical | C23×D4×C29 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C29 ◁ G | |
1968 conjugacy classes of subgroups; only the 952 normal subgroups are shown.
This group has 1968 conjugacy classes of subgroups — too many to draw. Only its 952 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 — 5 generators on 45 points:
(1,2)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(3,4);(6,8)(10,13)(11,15);(9,10)(11,14)(12,13)(15,16);(5,6)(7,8)(9,11,12,15)(10,14,13,16)1624 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#8304, C29×#8543, C29×#8548, … |
| maximal normal subgroup of | C29×#8304, C29×#8543, C29×#8548, … |
| maximal quotient of | C22×C29×#381, C22×C29×#385, C22×C29×#386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 2166);