Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2201 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23×C31 |
| prime radical | C31×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C22×C31 ◁ C2×D4×C31 ◁ G | |
226 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 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)(34,42)(35,39)(36,40)(37,41)(38,44)(43,59)(45,58)(46,57)(47,53)(48,54)(49,56)(50,60)(51,61)(52,62)(55,63);(32,34,37,45)(33,38,41,51)(35,46,48,43)(36,44,49,61)(39,52,54,50)(40,42,56,58)(47,62,63,60)(53,57,55,59);(32,35,37,48)(33,39,41,54)(34,43,45,46)(36,47,49,63)(38,50,51,52)(40,55,56,53)(42,57,58,59)(44,60,61,62)527 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#13679, C31×#13680, C2×C31×#2201, … |
| maximal normal subgroup of | C31×#13679, C31×#13680, C2×C31×#2201, … |
| maximal quotient of | C31×#12095, C31×#12101, C31×#12125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 933);