Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C23×C4 |
| direct product | C31×#3135 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| socle | C22×C31 | C22×D4 |
| prime radical | C31×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G | |
392 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 1984 |
A minimal faithful permutation representation — 4 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)(34,37)(35,36)(38,39)(40,41)(42,51)(43,47)(44,49)(45,54)(46,50)(48,53)(52,55);(33,36)(37,39)(40,42,45,43,46,53,44,52)(41,47,49,51,50,55,54,48);(32,34,35,38)(33,37,36,39)(40,43,45,53,46,52,44,42)(41,48,49,47,50,51,54,55);(40,44,46,45)(41,49,50,54)(42,52,53,43)(47,51,55,48)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#9005, C31×#9008, C31×#10033, … |
| maximal normal subgroup of | C31×#9005, C31×#9008, C31×#10033, … |
| maximal quotient of | C31×#20762, C31×#21086, C31×#21199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1867);