Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C25 |
| direct product | C2×C31×#461 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C31 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C22×D4×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G | |
544 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 544 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) = 45 |
| transitive degree | μt(G) = 1984 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(4,6)(7,8,10,13)(9,11,14,12);(3,4,5,6);(8,11)(9,14)(12,13)1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#8985, C31×#8988, C31×#9217, … |
| maximal normal subgroup of | C31×#8985, C31×#8988, C31×#9217, … |
| maximal quotient of | C2×C31×#1671, C2×C31×#1673, C2×C31×#1675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1762);