Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#1675 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C31 |
| prime radical | C2×D4×C31 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C42×C31 ◁ G | |
160 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 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)(32,33,35,38)(34,36,39,37)(40,41,44,47)(42,48,51,55)(43,52,49,46)(45,53,54,50);(40,42,49,53,44,51,43,50)(41,45,52,55,47,54,46,48);(33,36)(34,39)(37,38)899 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12372, C31×#12373, C31×#12374, … |
| maximal normal subgroup of | C31×#12372, C31×#12373, C31×#12374, … |
| maximal quotient of | C31×#8644, C31×#9215, C31×#9567, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 407);