Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#1813 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | #117 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C2×C31 | #380 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C2×M16×C31 ◁ G | |
252 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 generators on 63 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)(32,33,37,41,38,42,36,51)(34,43,46,54,47,58,45,39)(35,48,49,55,50,40,52,56)(44,59,60,63,61,57,62,53);(32,34,36,45,38,47,37,46)(33,39,41,43,42,54,51,58)(35,44,49,60,50,61,52,62)(40,53,55,57,48,63,56,59);(32,35)(33,40)(34,44)(36,49)(37,52)(38,50)(39,53)(41,55)(42,48)(43,57)(45,60)(46,62)(47,61)(51,56)(54,63)(58,59)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7227, C31×#7229, C31×#11870, … |
| maximal normal subgroup of | C31×#7227, C31×#7229, C31×#11870, … |
| maximal quotient of | C31×#8601, C31×#8602, C31×#8603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 545);