Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C26 |
| direct product | C22×C31×#100 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C31 | C22⋊C4 |
| commutator | C22 | C23×C4×C31 |
| Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C23×D4×C31 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G | |
992 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(3,4);(5,6,7,10,8,9,11,12);(6,9)(7,11)1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#10740, C31×#10749, C31×#10762, … |
| maximal normal subgroup of | C31×#10740, C31×#10749, C31×#10762, … |
| maximal quotient of | C22×C31×#323, C2×C31×#1460, C2×C31×#1462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1617);