Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#2214 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C31 |
| prime radical | C31×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C4×C8×C31 ◁ G | |
150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 generators on 63 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)(32,33,39,45)(34,43,48,54)(35,49,51,62)(36,44,38,42)(37,40,47,55)(41,56,58,46)(50,63,53,59)(52,60,61,57);(32,34,38,47,39,48,36,37)(33,40,44,54,45,55,42,43)(35,46,53,61,51,56,50,52)(41,49,60,63,58,62,57,59);(32,35)(33,41)(34,46)(36,50)(37,52)(38,53)(39,51)(40,49)(42,57)(43,59)(44,60)(45,58)(47,61)(48,56)(54,63)(55,62)713 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12312, C31×#12313, C31×#12315, … |
| maximal normal subgroup of | C31×#12312, C31×#12313, C31×#12315, … |
| maximal quotient of | C31×#12422, C31×#12423, C31×#12426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 946);