Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#3255 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C29 |
| prime radical | C23×C29 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C2×C42×C29 ◁ G | |
342 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 4 generators on 53 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)(38,39,43,46)(40,44,42,51)(41,48,50,52)(45,49,53,47);(30,31)(32,35)(33,34)(36,37)(38,40,43,42)(39,44,46,51)(41,49,50,47)(45,48,53,52);(31,34)(35,37)(38,41,43,50)(39,45,46,53)(40,47,42,49)(44,52,51,48);(30,32,33,36)(31,35,34,37)754 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#10113, C29×#10115, C29×#10117, … |
| maximal normal subgroup of | C29×#10113, C29×#10115, C29×#10117, … |
| maximal quotient of | C29×#21894, C29×#21964, C29×#22118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1987);