Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×C31×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C31 | C23 |
| commutator | C22 | C23×C4×C31 |
| Frattini, cosocle | C23 | C24×C31 |
| socle | C23×C31 | C24 |
| prime radical | C2×C31×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C31 ◁ G | |
924 conjugacy classes of subgroups; only the 368 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 368 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 1984 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(6,8)(10,11);(9,10,12,11);(5,6,7,8)(9,11,12,10)1736 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#7569, C31×#7587, C31×#7603, … |
| maximal normal subgroup of | C31×#7569, C31×#7587, C31×#7603, … |
| maximal quotient of | C4×C31×#390, C4×C31×#384, C4×C31×#392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1038);