Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C25 |
| direct product | C22×C29×SD32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | D8 |
| commutator, Frattini, cosocle | C8 | C24×C29 |
| prime radical | C22×D8×C29 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C22×C4×C29 ◁ C22×C8×C29 ◁ G | |
400 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 1856 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);(6,12)(7,19)(8,10)(9,17)(11,15)(14,20)(16,18)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#13013, C29×#13014, C29×#13018, … |
| maximal normal subgroup of | C29×#13013, C29×#13014, C29×#13018, … |
| maximal quotient of | C22×C29×#357, C22×C29×#358, C22×C29×#367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 2141);