Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C29 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C29×#99 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C22⋊C4 |
| commutator | C22 | C2×C42×C29 |
| Frattini, cosocle | C24 | C23×C29 |
| socle | C22×C29 | C2×C22⋊C4 |
| prime radical | C22×D4×C29 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C23×C4×C29 ◁ G | |
412 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(5,6)(7,11,12,9)(8,10);(5,7)(6,9)(8,12)(10,11)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#7204, C29×#7205, C29×#7282, … |
| maximal normal subgroup of | C29×#7204, C29×#7205, C29×#7282, … |
| maximal quotient of | C4×C29×#323, C4×C29×#324, C29×#8152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 486);