Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C22×C4 |
| direct product | C31×#1988 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C31 | C2×D4 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C22×C4 | C23×C31 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G | |
292 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 3968 |
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)(33,35)(36,37)(38,42)(39,40)(41,43)(44,45,46,49,47,50,51,48);(32,33,34,35)(37,40)(41,42)(45,48)(46,51)(49,50);(36,38,39,43)(37,41,40,42)1178 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C31×#8421, C31×#8427, C31×#8985, … |
| maximal normal subgroup of | C31×#8421, C31×#8427, C31×#8985, … |
| maximal quotient of | C31×#9865, C31×#9869, C31×#9898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 720);