Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#2048 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C2×C29×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C4×C29 ◁ G | |
274 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 80 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)(31,36)(32,39)(33,34)(37,44)(38,42)(40,41);(30,31)(32,37)(33,42)(34,36)(35,38)(39,43)(40,45)(41,44);(30,32,33,40,35,41,34,39)(31,37,36,43,38,44,42,45)667 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#2048, C29×#23525, C29×#24295, … |
| maximal normal subgroup of | C2×C29×#2048, C29×#23525, C29×#24295, … |
| maximal quotient of | C29×#10440, C29×#10441, C29×#10442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 780);