Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C31 · Schur multiplier (H2) = C25 |
| direct product | C31×#2913 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C31 | C22×D4 |
| commutator | C4 | C23×C4×C31 |
| Frattini, cosocle | C2×C4 | C24×C31 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C31 ◁ C2×#41×C31 ◁ G | |
476 conjugacy classes of subgroups; only the 344 normal subgroups are shown.
This group has 476 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) = 1984 |
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)(37,41)(39,42)(43,47)(44,48)(46,49)(50,51);(36,37,42,48,40,44,39,41)(38,43,49,51,45,50,46,47);(36,38)(37,43)(39,46)(40,45)(41,47)(42,49)(44,50)(48,51);(38,45)(43,50)(46,49)(47,51)1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#12925, C31×#12926, C31×#12996, … |
| maximal normal subgroup of | C31×#12925, C31×#12926, C31×#12996, … |
| maximal quotient of | C31×#17789, C31×#17799, C31×#17812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1645);