Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C25 |
| direct product | C29×#1784 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C24×C29 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C23×C4×C29 ◁ G | |
564 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 49 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)(34,35,37,41,38,40,46,48)(36,42,43,49,44,39,45,47);(34,36,38,44)(35,39,40,42)(37,45,46,43)(41,49,48,47);(31,33)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#7308, C29×#7315, C29×#7317, … |
| maximal normal subgroup of | C29×#7308, C29×#7315, C29×#7317, … |
| maximal quotient of | C29×#8299, C29×#8303, C29×#8306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 516);