Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#2015 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C31 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C22×D4×C31 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G | |
390 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 390 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) = 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)(34,35)(40,41,45,49)(42,50,53,54)(43,48,44,47)(46,52,55,51);(33,35)(36,37)(38,39)(40,42,45,53)(41,46,49,55)(43,52,44,51)(47,54,48,50);(37,39)(41,47)(43,44)(46,54)(48,49)(50,55)(51,52)992 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C31×#2015, C31×#23421, C31×#23423, … |
| maximal normal subgroup of | C2×C31×#2015, C31×#23421, C31×#23423, … |
| maximal quotient of | C31×#9955, C31×#9956, C31×#9958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 747);