Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C29 · Schur multiplier (H2) = C22 |
| direct product | C8×SD16×C29 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C29 | D4 |
| commutator | C4 | C22×C8×C29 |
| Frattini, cosocle | C42 | C23×C29 |
| socle | C22×C29 | C4×D4 |
| prime radical | C2×D4×C29 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C29 ◁ C4×C8×C29 ◁ G | |
180 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(9,10,11,12,13,14,15,16);(10,12)(11,15)(14,16)1624 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C16×SD16×C29, C29×#12049, C29×#12050, … |
| maximal normal subgroup of | C16×SD16×C29, C29×#12049, C29×#12050, … |
| maximal quotient of | C8×C29×#102, C8×C29×#103, C8×C29×#106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 308);