Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C23×C4 |
| direct product | C29×#3131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C29 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C29 |
| socle | C22×C29 | C22×D4 |
| prime radical | C29×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C29 ◁ C2×C42×C29 ◁ G | |
400 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1856 |
A minimal faithful permutation representation — 4 generators on 53 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)(30,31)(32,35)(33,34)(36,37)(38,39)(40,46)(41,45)(42,47)(43,48)(44,49)(50,53)(51,52);(31,34)(35,37)(38,40,43,50,44,51,42,41)(39,45,47,52,49,53,48,46);(30,32,33,36)(31,35,34,37)(38,41,42,51,44,50,43,40)(39,46,48,53,49,52,47,45);(38,42,44,43)(39,47,49,48)(40,41,51,50)(45,52,53,46)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#9067, C29×#9070, C29×#9751, … |
| maximal normal subgroup of | C29×#9067, C29×#9070, C29×#9751, … |
| maximal quotient of | C29×#20760, C29×#21083, C29×#21195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1863);