Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#1912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #386 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C22×C4×C31 ◁ G | |
314 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 314 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)(32,33,37,41,38,42,36,52)(34,43,47,54,46,58,45,39)(35,48,50,56,51,40,49,55)(44,57,61,53,60,63,59,62);(32,34)(33,39,42,54)(35,44)(36,45)(37,47)(38,46)(40,53,48,62)(41,43,52,58)(49,59)(50,61)(51,60)(55,63,56,57);(32,35,38,51)(33,40,42,48)(34,44,46,60)(36,50,37,49)(39,53,54,62)(41,56,52,55)(43,57,58,63)(45,61,47,59)899 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12781, C31×#12782, C2×C31×#1912, … |
| maximal normal subgroup of | C31×#12781, C31×#12782, C2×C31×#1912, … |
| maximal quotient of | C31×#9100, C31×#9102, C31×#9104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 644);