Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C24 |
| direct product | C31×#3160 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C31 |
| socle | C22×C31 | C22×D4 |
| prime radical | C23×C31 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G | |
356 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1984 |
A minimal faithful permutation representation — 4 generators on 55 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)(32,33,35,37)(34,38,39,36)(40,41)(42,49)(43,47)(44,53)(45,48)(46,51)(50,54)(52,55);(32,34,35,39)(33,36,37,38);(40,42,44,51,45,52,43,50)(41,46,47,49,48,54,53,55);(40,43,45,44)(41,47,48,53)(42,50,52,51)(46,49,54,55)1364 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#9068, C31×#9749, C31×#9752, … |
| maximal normal subgroup of | C31×#9068, C31×#9749, C31×#9752, … |
| maximal quotient of | C31×#20797, C31×#21151, C31×#21318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 1892);