Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#1904 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #386 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C2×C29×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C4×C29 ◁ G | |
346 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 464 |
A minimal faithful permutation representation — 3 generators on 45 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)(30,31,34,37,35,39,33,38)(32,40,42,44,43,45,41,36);(30,32)(31,36,39,44)(33,41)(34,42)(35,43)(37,40,38,45);(31,37)(33,34)(36,40)(38,39)(41,42)(44,45)841 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1904, C29×#20998, C29×#22284, … |
| maximal normal subgroup of | C2×C29×#1904, C29×#20998, C29×#22284, … |
| maximal quotient of | C29×#9087, C29×#9089, C29×#9091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 636);