Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C2 |
| direct product | C29×#2237 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C29 |
| prime radical | C23×C29 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ C22×C8×C29 ◁ G | |
132 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 69 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)(33,34)(36,37)(38,39,43,48)(40,44,51,59)(41,45,54,61)(42,55,53,47)(46,62,60,52)(49,56,65,68)(50,66,64,58)(57,69,67,63);(31,34)(35,37)(38,40,43,51)(39,44,48,59)(41,49,54,65)(42,50,53,64)(45,56,61,68)(46,57,60,67)(47,58,55,66)(52,63,62,69);(30,32,33,36)(31,35,34,37)(39,45)(41,52)(42,53)(44,56)(46,47)(48,61)(49,63)(50,64)(54,62)(55,60)(57,58)(59,68)(65,69)(66,67)841 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#2237, C29×#33645, C29×#33647, … |
| maximal normal subgroup of | C2×C29×#2237, C29×#33645, C29×#33647, … |
| maximal quotient of | C29×#12868, C29×#12921, C29×#12994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 969);