Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C24 |
| direct product | C2×C31×#477 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C31 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G | |
416 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 416 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) = 49 |
| transitive degree | μt(G) = 3968 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(3,4,6,8)(5,9,10,7)(12,15)(13,16)(17,18);(3,5,6,10)(4,7,8,9)(11,12,14,15)(13,16,17,18);(11,13,14,17)(12,16,15,18)1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#8974, C31×#8975, C31×#9216, … |
| maximal normal subgroup of | C31×#8974, C31×#8975, C31×#9216, … |
| maximal quotient of | C2×C31×#1803, C2×C31×#1863, C2×C31×#1921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1806);