Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C29 · Schur multiplier (H2) = C29 |
| direct product | C2×C29×#379 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C29 | C23 |
| commutator | C22 | C23×C4×C29 |
| Frattini, cosocle | C23 | C24×C29 |
| prime radical | C26×C29 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C29 ◁ G | |
3464 conjugacy classes of subgroups; only the 600 normal subgroups are shown.
This group has 3464 conjugacy classes of subgroups — too many to draw. Only its 600 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(4,6)(7,8,9,10)(12,14);(11,12)(13,14);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)1624 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C29×#1304, C29×#8101, C29×#10725, … |
| maximal normal subgroup of | C2×C29×#1304, C29×#8101, C29×#10725, … |
| maximal quotient of | C2×C29×C22⋊#95, C4×C29×#379, C2×C29×#1438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1009);