Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2025 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C31 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| prime radical | C22×D4×C31 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G | |
294 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 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)(36,37,39,41)(38,42,43,40)(44,45,46,48)(47,49,51,50);(32,33)(34,35)(36,38,39,43)(37,40,41,42)(45,48)(49,50);(33,35)(45,49)(47,51)(48,50)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#7428, C2×C31×#2025, C31×#23451, … |
| maximal normal subgroup of | C31×#7428, C2×C31×#2025, C31×#23451, … |
| maximal quotient of | C31×#10344, C31×#10346, C31×#10351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 757);