Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C29×#353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #99 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C2×D4 | C23×C29 |
| prime radical | C2×C29×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C23×C29 ◁ C22×D4×C29 ◁ G | |
368 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 464 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(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);(3,4)(5,8,10,7)(6,9);(4,7)(5,10)(8,9)754 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#7485, C29×#7493, C29×#7498, … |
| maximal normal subgroup of | C29×#7485, C29×#7493, C29×#7498, … |
| maximal quotient of | C2×C29×#1318, C2×C29×#1325, C29×#11475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 856);