Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C26 |
| direct product | C31×#2889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | C22×C22⋊C4 |
| commutator | C22 | C23×C4×C31 |
| Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C2×C31×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×#41×C31 ◁ G | |
704 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 340 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 — 4 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)(32,33,36,41,37,40,49,54)(34,42,44,55,45,38,48,51)(35,39,47,53,46,52,59,62)(43,56,58,63,57,50,60,61);(32,34,37,45)(33,38,40,42)(35,43,46,57)(36,48,49,44)(39,50,52,56)(41,55,54,51)(47,60,59,58)(53,63,62,61);(32,35)(33,39)(34,43)(36,47)(37,46)(38,50)(40,52)(41,53)(42,56)(44,58)(45,57)(48,60)(49,59)(51,61)(54,62)(55,63);(35,46)(39,52)(43,57)(47,59)(50,56)(53,62)(58,60)(61,63)1271 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#13096, C31×#13098, C31×#21780, … |
| maximal normal subgroup of | C31×#13096, C31×#13098, C31×#21780, … |
| maximal quotient of | C31×#17351, C31×#17352, C31×#17368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1621);