Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#2024 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C29 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C29 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C2×C42×C29 ◁ G | |
290 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 3712 |
A minimal faithful permutation representation — 3 generators on 49 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)(34,35,37,39)(36,40,41,38)(43,46)(44,48)(47,49);(30,31)(32,33)(34,36,37,41)(35,38,39,40);(31,33)(42,43,45,47)(44,49,48,46)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#7753, C29×#7755, C29×#7759, … |
| maximal normal subgroup of | C29×#7753, C29×#7755, C29×#7759, … |
| maximal quotient of | C29×#10341, C29×#10343, C29×#10349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 756);