Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22 |
| direct product | C31×#1952 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | #121 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C23×C31 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C22×C4×C31 ◁ G | |
196 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 196 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) = 51 |
| transitive degree | μt(G) = 1984 |
A minimal faithful permutation representation — 3 generators on 51 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)(32,33,35,34)(37,42)(38,45)(39,40)(43,50)(44,48)(46,47);(32,34,35,33)(36,37)(38,43)(39,48)(40,42)(41,44)(45,49)(46,51)(47,50);(36,38,39,46,41,47,40,45)(37,43,48,51,44,50,42,49)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12940, C31×#12941, C31×#12945, … |
| maximal normal subgroup of | C31×#12940, C31×#12941, C31×#12945, … |
| maximal quotient of | C31×#9608, C31×#9610, C31×#9613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 684);