Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22×C4 |
| direct product | C31×#1996 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C31 |
| Frattini, cosocle | C24 | C23×C31 |
| prime radical | C24×C31 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G | |
324 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 generators on 47 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)(32,33)(34,38)(35,36)(37,39)(40,41)(42,44,46,47)(43,45);(33,36)(37,38)(40,42)(41,44,45,47)(43,46);(32,34,35,39)(33,37,36,38)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#8467, C31×#8468, C31×#8480, … |
| maximal normal subgroup of | C31×#8467, C31×#8468, C31×#8480, … |
| maximal quotient of | C31×#9878, C31×#9879, C31×#9880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 728);