Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | C2×#99 |
| commutator | C2×C4 | C22×C4×C31 |
| Frattini, cosocle | C2×Q8 | C23×C31 |
| prime radical | C2×#41×C31 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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)(32,33,36,40,37,41,52,59)(34,42,45,60,43,56,50,62)(35,46,48,58,49,39,51,57)(38,53,55,47,54,61,63,44);(32,34)(33,38)(35,47)(36,50)(37,43)(39,56)(40,55)(41,54)(42,46)(44,49)(45,52)(48,53)(51,61)(57,62)(58,60)(59,63);(34,43)(38,54)(42,56)(44,47)(45,50)(53,61)(55,63)(60,62)713 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12709, C31×#12710, C31×#12729, … |
| maximal normal subgroup of | C31×#12709, C31×#12710, C31×#12729, … |
| maximal quotient of | C31×#11432, C31×#11448, C31×#11480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 865);