Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C31 · Schur multiplier (H2) = C25 |
| direct product | C31×#3416 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C31 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ #41×C31 ◁ C31×#131 ◁ G | |
360 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 992 |
A minimal faithful permutation representation — 4 generators on 63 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)(32,33,37,42)(34,39,43,53)(35,38,48,52)(36,49,47,41)(40,56,51,46)(44,50,57,61)(45,60,58,55)(54,63,62,59);(33,38)(35,46)(36,47)(39,50)(40,41)(42,52)(44,59)(45,58)(48,56)(49,51)(53,61)(54,55)(57,63)(60,62);(32,34)(33,39)(35,44)(36,45)(37,43)(38,50)(40,54)(41,55)(42,53)(46,59)(47,58)(48,57)(49,60)(51,62)(52,61)(56,63);(34,43)(39,53)(44,57)(45,58)(50,61)(54,62)(55,60)(59,63)992 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#33854, C31×#33855, C31×#33856, … |
| maximal normal subgroup of | C31×#33854, C31×#33855, C31×#33856, … |
| maximal quotient of | C31×#33381, C31×#33382, C31×#33411, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 2148);