Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C2 |
| direct product | C31×#2231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C23×C31 |
| prime radical | C31×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C4×C8×C31 ◁ G | |
120 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 60 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)(32,33,38,43)(34,44,47,42)(35,48)(36,52,45,41)(37,39,54,56)(40,57)(46,60)(49,62)(50,61)(51,55)(53,58)(59,63);(32,34,45,54,38,47,36,37)(33,39,52,42,43,56,41,44)(35,46,50,63,51,57,49,53)(40,55,59,61,60,48,58,62);(32,35)(33,40)(34,46)(36,49)(37,53)(38,51)(39,55)(41,58)(42,61)(43,60)(44,62)(45,50)(47,57)(48,56)(52,59)(54,63)713 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12463, C31×#12464, C31×#12465, … |
| maximal normal subgroup of | C31×#12463, C31×#12464, C31×#12465, … |
| maximal quotient of | C31×#12824, C31×#12950, C31×#13116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 963);