Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C23×C4 |
| direct product | C29×#1875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C2×C29×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C23×C4×C29 ◁ G | |
412 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 3 generators on 45 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)(31,33)(34,35)(36,37)(38,39,43,44,42,40,41,45);(30,31,32,33)(38,40,42,39)(41,45,43,44);(35,37)(38,41,42,43)(39,44,40,45)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#8487, C2×C29×#1875, C29×#20908, … |
| maximal normal subgroup of | C29×#8487, C2×C29×#1875, C29×#20908, … |
| maximal quotient of | C29×#8940, C29×#8943, C29×#8978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 607);