Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C4 |
| direct product | C31×#1922 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C22×C31 | C4×D4 |
| prime radical | C23×C31 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C22×C8×C31 ◁ G | |
186 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 992 |
A minimal faithful permutation representation — 3 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)(34,37)(35,36)(38,39)(40,41,44,48,45,43,52,47)(42,46,50,54,51,49,55,53);(33,36)(37,39)(40,42,44,50,45,51,52,55)(41,46,48,54,43,49,47,53);(32,34,35,38)(33,37,36,39)(40,43)(41,45)(42,49)(44,47)(46,51)(48,52)(50,53)(54,55)1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C31×#1922, C31×#21041, C31×#21042, … |
| maximal normal subgroup of | C2×C31×#1922, C31×#21041, C31×#21042, … |
| maximal quotient of | C31×#9533, C31×#9534, C31×#9535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 654);