Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C2×C4 |
| direct product | C31×#1833 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | #117 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C2×C31 | #382 |
| prime radical | C23×C31 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C31×#117 ◁ G | |
172 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 100 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)(34,42,45,38)(35,46,48,61)(36,41)(37,40)(39,47,52,62)(43,56)(44,55,59,51)(49,53)(50,57)(54,63)(58,60);(32,34,36,44,37,45,49,59)(33,38,41,51,40,42,53,55)(35,43,47,57,48,58,62,63)(39,50,46,60,52,54,61,56);(32,35)(33,39,40,52)(34,43)(36,47)(37,48)(38,50,42,54)(41,46,53,61)(44,57)(45,58)(49,62)(51,60,55,56)(59,63)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12826, C31×#12827, C31×#12828, … |
| maximal normal subgroup of | C31×#12826, C31×#12827, C31×#12828, … |
| maximal quotient of | C31×#8780, C31×#8781, C31×#8782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 565);