Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C22 |
| direct product | C29×#1946 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| socle | C22×C29 | C4×D4 |
| prime radical | C23×C29 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C22×C8×C29 ◁ G | |
200 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 928 |
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)(30,31)(32,33)(34,35,38,43,39,44,48,42)(36,40,37,46,45,41,47,49);(31,33)(34,36,38,37,39,45,48,47)(35,40,43,46,44,41,42,49);(34,37,38,45,39,47,48,36)(35,41,42,46,44,40,43,49)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#7125, C29×#12944, C29×#12956, … |
| maximal normal subgroup of | C29×#7125, C29×#12944, C29×#12956, … |
| maximal quotient of | C29×#9595, C29×#9596, C29×#9597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 678);