Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C2 |
| direct product | C31×#2239 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C23×C31 |
| prime radical | C23×C31 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×M16×C31 ◁ G | |
124 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 56 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)(32,33,37,41)(34,42,43,60)(35,46,47,63)(36,51,49,40)(38,53,54,44)(39,56,57,50)(45,58,52,61)(48,55,62,59);(35,47)(39,57)(42,60)(44,53)(46,63)(48,62)(50,56)(58,61);(32,34,36,52,37,43,49,45)(33,38,40,59,41,54,51,55)(35,48,50,53,47,62,56,44)(39,58,46,60,57,61,63,42)620 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12299, C31×#12300, C31×#12302, … |
| maximal normal subgroup of | C31×#12299, C31×#12300, C31×#12302, … |
| maximal quotient of | C31×#12875, C31×#12931, C31×#13009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 971);