Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#2007 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C31×#537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C22×C4×C31 ◁ G | |
370 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 992 |
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)(36,37,41,45)(38,46,48,51)(39,44,40,43)(42,49,50,47);(32,33)(34,35)(36,38)(37,42)(39,49)(40,47)(41,48)(43,51)(44,46)(45,50);(33,35)(36,38)(37,42)(39,47)(40,49)(41,48)(43,46)(44,51)(45,50)806 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7409, C31×#12098, C31×#12100, … |
| maximal normal subgroup of | C31×#7409, C31×#12098, C31×#12100, … |
| maximal quotient of | C31×#9934, C31×#9935, C31×#9936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 739);