Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C2×C29×#331 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C22⋊C4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C42 | C23×C29 |
| prime radical | C2×C29×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C2×C42×C29 ◁ G | |
360 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47);(3,4,5,8,6,7,9,10)(11,12)(13,15)(14,17)(16,18);(4,7)(5,9)(12,15)(13,18)(16,17)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#7485, C4×C29×#331, C29×#8231, … |
| maximal normal subgroup of | C29×#7485, C4×C29×#331, C29×#8231, … |
| maximal quotient of | C2×C29×#1270, C29×#7569, C29×#7640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 270);