Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#1915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #386 |
| commutator | C23 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G | |
308 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 108 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)(32,33,36,41,37,40,42,55)(34,38,45,52,44,53,49,56)(35,46,48,57,47,39,50,54)(43,51,59,62,58,61,60,63);(32,34,42,45,37,44,36,49)(33,38,41,56,40,53,55,52)(35,43,48,60,47,58,50,59)(39,51,57,62,46,61,54,63);(32,35)(33,39)(34,43)(36,50)(37,47)(38,51)(40,46)(41,57)(42,48)(44,58)(45,60)(49,59)(52,63)(53,61)(54,55)(56,62)899 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12757, C31×#12758, C31×#12770, … |
| maximal normal subgroup of | C31×#12757, C31×#12758, C31×#12770, … |
| maximal quotient of | C31×#9108, C31×#9110, C31×#9189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 647);