Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#1883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #120 |
| 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 ◁ C22×C29 ◁ C22×C4×C29 ◁ G | |
370 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 41 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)(31,33)(34,35,37,39,38,36,41,40);(30,31,32,33)(34,36)(35,38)(37,39)(40,41);(34,35)(36,38)(37,39)(40,41)928 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1883, C29×#20942, C29×#20943, … |
| maximal normal subgroup of | C2×C29×#1883, C29×#20942, C29×#20943, … |
| maximal quotient of | C29×#8954, C29×#8956, C29×#8969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 615);