Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C2 |
| direct product | C31×#2225 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C31 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C31 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C42×C31 ◁ C4×C8×C31 ◁ G | |
130 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 3968 |
A minimal faithful permutation representation — 3 generators on 55 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)(32,33,35,37)(34,38,39,36)(41,45)(42,48)(43,50)(46,55)(47,54)(49,52)(51,53);(32,34,35,39)(33,36,37,38);(40,41,44,47)(42,49,51,55)(43,52,50,46)(45,53,54,48)899 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12369, C31×#12370, C31×#12377, … |
| maximal normal subgroup of | C31×#12369, C31×#12370, C31×#12377, … |
| maximal quotient of | C31×#12812, C31×#12938, C31×#13110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 957);