Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C25 |
| direct product | C31×#2911 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator | C4 | C23×C4×C31 |
| Frattini, cosocle | C2×C4 | C24×C31 |
| socle | C22×C31 | C22×D4 |
| prime radical | C24×C31 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C23×C4×C31 ◁ G | |
492 conjugacy classes of subgroups; only the 344 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 344 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 — 4 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)(32,33,34,35)(36,37)(38,45)(39,43)(40,49)(41,44)(42,48)(46,51)(47,50);(36,38,40,48,41,46,39,47)(37,42,43,45,44,50,49,51);(38,46)(42,50)(45,51)(47,48);(36,39,41,40)(37,43,44,49)(38,47,46,48)(42,45,50,51)1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#10895, C31×#10900, C31×#11100, … |
| maximal normal subgroup of | C31×#10895, C31×#10900, C31×#11100, … |
| maximal quotient of | C31×#17787, C31×#17795, C31×#17797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1643);