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×#1496 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C31 | C2×D4 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C42 | C23×C31 |
| socle | C22×C31 | C2×C22⋊C4 |
| prime radical | C31×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G | |
246 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 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)(34,38)(35,37)(36,39)(40,41,43,47,44,46,50,53)(42,45,49,51,48,52,54,55);(33,37)(34,35)(38,39)(40,42)(41,45)(43,49)(44,48)(46,52)(47,51)(50,54)(53,55);(32,34,36,35)(33,38,39,37)(41,46)(42,48)(47,53)(49,54)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#8489, C31×#8500, C31×#8503, … |
| maximal normal subgroup of | C31×#8489, C31×#8500, C31×#8503, … |
| maximal quotient of | C31×#7698, C31×#7699, C31×#7700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 228);