Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#1404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C29 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C4×C29 |
| Frattini, cosocle | #124 | C22×C29 |
| prime radical | C29×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C29 ◁ C22×C29 ◁ C2×C4×C29 ◁ C29×#124 ◁ G | |
112 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 28 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 — 2 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)(30,31,32,36)(33,42,43,39)(34,38,41,45)(35,44,37,40);(30,31)(32,39)(33,42)(34,38)(35,41)(36,40)(37,44)(43,45)493 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1404, C29×#13568, C29×#13569, … |
| maximal normal subgroup of | C2×C29×#1404, C29×#13568, C29×#13569, … |
| maximal quotient of | C29×#7345, C29×#7376, C29×#7408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 136);