Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#1872 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C2×D4 |
| commutator | C2×C4 | C22×C4×C29 |
| Frattini, cosocle | C22×C4 | C23×C29 |
| prime radical | C24×C29 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C23×C4×C29 ◁ G | |
404 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 144 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)(38,39,41,43,42,45,40,44);(30,31,32,33)(39,43)(40,41)(44,45);(35,37)1102 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#8405, C29×#8406, C29×#8407, … |
| maximal normal subgroup of | C29×#8405, C29×#8406, C29×#8407, … |
| maximal quotient of | C29×#8937, C29×#8942, C29×#8973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 604);