Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C29 · Schur multiplier (H2) = C26 |
| direct product | C29×#1779 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C29 | C23 |
| commutator | C22 | C22×C8×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C23×C29 | C22×C4 |
| prime radical | C25×C29 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C29 ◁ G | |
852 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 208 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) = 928 |
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)(32,33)(34,35)(36,37)(38,39,40,42,41,43,44,45);(31,33);(35,37)1624 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#7032, C29×#7035, C29×#7047, … |
| maximal normal subgroup of | C29×#7032, C29×#7035, C29×#7047, … |
| maximal quotient of | C29×#8298, C29×#8299, C29×#8300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 511);