Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C2×C4 |
| direct product | C31×#1929 |
| 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 | C2×#41×C31 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C22×C8×C31 ◁ G | |
224 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 224 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) = 51 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 3 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)(33,34)(36,37,40,44,41,39,48,43)(38,42,46,50,47,45,51,49);(32,33,35,34)(36,38,40,46,41,47,48,51)(37,42,44,50,39,45,43,49);(32,34,35,33)(36,39)(37,41)(38,45)(40,43)(42,47)(44,48)(46,49)(50,51)1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C31×#1929, C31×#21070, C31×#21071, … |
| maximal normal subgroup of | C2×C31×#1929, C31×#21070, C31×#21071, … |
| maximal quotient of | C31×#9541, C31×#9542, C31×#9543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 661);