Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#1622 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C29 |
| prime radical | C29×#459 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C42×C29 ◁ G | |
236 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 1856 |
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)(31,34)(32,36)(35,37)(39,42)(40,44)(43,45);(30,31,33,35)(32,37,36,34)(40,44)(42,45);(38,39)(40,45)(41,43)(42,44)754 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1622, C29×#18480, C29×#18518, … |
| maximal normal subgroup of | C2×C29×#1622, C29×#18480, C29×#18518, … |
| maximal quotient of | C29×#7735, C29×#7745, C29×#7747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 354);