Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C2×C4 |
| direct product | C31×#1995 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C22×C31 | C2×C22⋊C4 |
| prime radical | C23×C31 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G | |
244 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 generators on 55 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)(34,38)(35,36)(37,39)(40,41,44,47,45,48,43,53)(42,49,52,54,50,55,51,46);(33,36)(37,38)(40,42)(41,46,48,54)(43,51)(44,52)(45,50)(47,49,53,55);(32,34,35,39)(33,37,36,38)(42,50)(46,54)(49,55)(51,52)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7129, C31×#7360, C31×#8494, … |
| maximal normal subgroup of | C31×#7129, C31×#7360, C31×#8494, … |
| maximal quotient of | C31×#9872, C31×#9873, C31×#9874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 727);