Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C26 |
| direct product | C2×C29×#455 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C29 |
| prime radical | C2×C29×H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ G | |
760 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47);(3,4,7,10)(5,11,13,18)(6,14,16,9)(8,15,17,12);(3,5)(4,8)(6,15)(7,13)(9,18)(10,17)(11,14)(12,16);(6,16)(9,14)(11,18)(12,15)928 conjugacy classes — view the full character table.
| maximal subgroup of | C29×#13048, C29×#13050, C29×#13348, … |
| maximal normal subgroup of | C29×#13048, C29×#13050, C29×#13348, … |
| maximal quotient of | C2×C29×#1601, C2×C29×#1602, C2×C29×#1607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1747);