Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C23 |
| direct product | C2×C31×#352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | #99 |
| commutator | C23 | C22×C4×C31 |
| Frattini, cosocle | C2×D4 | C23×C31 |
| prime radical | C22×D4×C31 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C23×C31 ◁ C22×D4×C31 ◁ G | |
278 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(3,4)(5,11)(6,14,15,9)(7,10)(8,12,16,18)(13,17);(3,5,7,13)(4,8,10,16)(6,12,15,18)(9,11,14,17)806 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C31×#352, C31×#12621, C31×#12688, … |
| maximal normal subgroup of | C4×C31×#352, C31×#12621, C31×#12688, … |
| maximal quotient of | C2×C31×#1317, C2×C31×#1326, C31×#11472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 851);