Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C27 |
| direct product | C2×C29×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #120 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C24×C29 ◁ G | |
1018 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 464 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(7,10);(3,4)(5,7)(6,8)(9,10);(3,5)(4,7)(6,9)(8,10)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#13041, C29×#13043, C29×#13343, … |
| maximal normal subgroup of | C29×#13041, C29×#13043, C29×#13343, … |
| maximal quotient of | C2×C29×#1595, C2×C29×#1601, C2×C29×#1596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1755);