Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C27 |
| direct product | C2×C29×#393 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C29 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| prime radical | C25×C29 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C29 ◁ G | |
1036 conjugacy classes of subgroups; only the 328 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 328 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) = 1856 |
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);(8,10)(11,12,14,16)(13,17,18,15);(4,6)(8,10)(11,13,14,18)(12,15,16,17);(3,4)(5,6)(7,8)(9,10)(11,12)(13,17)(14,16)(15,18)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#9013, C29×#9016, C29×#9916, … |
| maximal normal subgroup of | C29×#9013, C29×#9016, C29×#9916, … |
| maximal quotient of | C2×C29×#1439, C2×C29×#1442, C2×C29×#1446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1117);