Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C29 · Schur multiplier (H2) = C25 |
| direct product | C22×C29×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C29 | D4 |
| commutator | C4 | C23×C4×C29 |
| Frattini, cosocle | C2×C4 | C24×C29 |
| socle | C23×C29 | C4⋊C4 |
| prime radical | C24×C29 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C29 ◁ C23×C4×C29 ◁ G | |
480 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 360 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);(5,6,9,12,10,13,8,18)(7,14,16,19,17,20,15,11);(5,7,8,15,10,17,9,16)(6,11,12,14,13,19,18,20)1624 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#10859, C29×#10860, C29×#10864, … |
| maximal normal subgroup of | C29×#10859, C29×#10860, C29×#10864, … |
| maximal quotient of | C22×C29×#334, C22×C29×#335, C2×C29×#1564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1646);