Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C29×#486 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C29 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C29 |
| prime radical | C2×C29×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ C2×C42×C29 ◁ G | |
528 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 232 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);(3,4)(5,8)(6,7)(9,10)(12,17)(13,18)(14,15);(4,7)(8,10)(11,12,15,13,16,18,14,17);(3,5,6,9)(4,8,7,10)(11,13,14,12,16,17,15,18)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#9057, C29×#9062, C29×#9735, … |
| maximal normal subgroup of | C29×#9057, C29×#9062, C29×#9735, … |
| maximal quotient of | C2×C29×#1834, C2×C29×#1934, C2×C29×#1957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1860);