Finite Group DatabasebetaFinite Group Databasebeta| order | 3712 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C29 · Schur multiplier (H2) = C24 |
| direct product | C29×#2859 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C29 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C29 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C29 ◁ G | |
472 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 3712 |
A minimal faithful permutation representation — 4 generators on 49 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)(36,39)(37,41)(42,43,47,46)(44,49,48,45);(34,35,38,40)(36,37,39,41)(42,44)(43,45)(46,49)(47,48);(30,31,32,33)(34,36,38,39)(35,37,40,41)(42,45,47,49)(43,48,46,44);(34,37)(35,39)(36,40)(38,41)(42,46,47,43)(44,49,48,45)1276 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C29×#7660, C29×#7663, C29×#7680, … |
| maximal normal subgroup of | C29×#7660, C29×#7663, C29×#7680, … |
| maximal quotient of | C29×#14687, C29×#14699, C29×#14707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3712, 1591);