Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C2×C4 |
| direct product | C29×#1931 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #121 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C22×D4×C29 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ G | |
232 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 96 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)(30,31,33,32)(35,39)(36,43)(37,40)(41,48)(42,46)(45,49);(30,32,33,31)(34,35,40,46,38,42,37,39)(36,41,47,48,43,49,44,45);(34,36,37,44,38,43,40,47)(35,41,39,45,42,49,46,48)928 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C29×#1931, C29×#21075, C29×#21076, … |
| maximal normal subgroup of | C2×C29×#1931, C29×#21075, C29×#21076, … |
| maximal quotient of | C29×#9550, C29×#9552, C29×#9563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 663);