Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C29 · Schur multiplier (H2) = C29 |
| direct product | C4×C29×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C24 |
| commutator | C2 | C24×C4×C29 |
| Frattini, cosocle | C22 | C25×C29 |
| socle | C22×C29 | C25 |
| prime radical | C2×C29×2−1+4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ G | |
1492 conjugacy classes of subgroups; only the 1372 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 1372 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) = 1856 |
A minimal faithful permutation representation — 5 generators on 49 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(6,10)(8,16)(11,13)(14,20);(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)1972 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#8319, C29×#17439, C29×#17454, … |
| maximal normal subgroup of | C29×#8319, C29×#17439, C29×#17454, … |
| maximal quotient of | C4×C29×#520, C4×C29×#519, C4×C29×#536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 2162);