Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#2074 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C31 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G | |
222 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 3968 |
A minimal faithful permutation representation — 3 generators on 59 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)(32,33,34,35)(36,37)(38,41)(39,40)(42,43)(44,45)(46,49)(47,50)(48,51)(52,59)(53,57)(54,56)(55,58);(37,40)(41,43)(44,46,47,53)(45,49,50,57)(48,52,55,54)(51,59,58,56);(33,35)(36,38,39,42)(37,41,40,43)(45,50)(46,52)(48,55)(49,56)(53,54)(57,59)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7413, C2×C31×#2074, C31×#23627, … |
| maximal normal subgroup of | C31×#7413, C2×C31×#2074, C31×#23627, … |
| maximal quotient of | C31×#10623, C31×#10624, C31×#10626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 806);