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×#2238 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C23×C31 |
| prime radical | C31×#469 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×M16×C31 ◁ G | |
134 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 134 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) = 47 |
| transitive degree | μt(G) = 496 |
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)(33,37)(34,40)(35,43)(38,45)(39,42)(41,47)(44,46);(34,41)(37,45)(40,47)(42,46);(32,33,35,44,36,38,43,39)(34,42,40,45,41,46,47,37)620 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12298, C31×#12300, C31×#12301, … |
| maximal normal subgroup of | C31×#12298, C31×#12300, C31×#12301, … |
| maximal quotient of | C31×#12875, C31×#12931, C31×#13010, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 970);