Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#2987 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C24 |
| commutator | C22 | C23×C4×C31 |
| Frattini, cosocle | C2×C4 | C24×C31 |
| socle | C22×C31 | C23×C4 |
| prime radical | C23×C31 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ G | |
314 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 248 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) = 1984 |
A minimal faithful permutation representation — 4 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)(34,40)(35,38)(36,39)(37,42)(41,46)(43,45)(44,47)(48,49,51,55,52,54,58,61)(50,53,57,60,56,59,62,63);(32,34,36,42)(33,37,39,40)(35,43,44,41)(38,46,47,45)(48,50)(49,53)(51,57)(52,56)(54,59)(55,60)(58,62)(61,63);(32,35,36,44)(33,38,39,47)(34,41,42,43)(37,45,40,46)(49,54)(50,56)(55,61)(57,62);(33,39)(37,40)(38,47)(45,46)(50,56)(53,59)(57,62)(60,63)1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#8919, C31×#9006, C31×#9008, … |
| maximal normal subgroup of | C31×#8919, C31×#9006, C31×#9008, … |
| maximal quotient of | C31×#20539, C31×#20542, C31×#20576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1719);