Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#1906 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C29 | #121 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C2×C29 | #386 |
| prime radical | C29×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C29 ◁ C22×C4×C29 ◁ G | |
332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 332 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)(32,35,39,45)(33,38)(34,37)(36,40,44,41)(42,43);(30,32)(31,35,37,45)(33,41)(34,39)(36,43,44,38)(40,42);(30,32)(31,36)(33,41)(34,39)(35,43)(37,44)(38,45)(40,42)928 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1906, C29×#21000, C29×#21558, … |
| maximal normal subgroup of | C2×C29×#1906, C29×#21000, C29×#21558, … |
| maximal quotient of | C29×#9095, C29×#9096, C29×#9097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 638);