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×#2135 |
| 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 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×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)(32,33,37,42,38,43,35,40)(34,44,47,62,48,59,45,61)(36,50,49,60,52,41,53,58)(39,54,55,51,56,63,57,46);(32,34,38,48)(33,39,43,56)(35,47,37,45)(36,51,52,46)(40,57,42,55)(41,59,50,44)(49,54,53,63)(58,61,60,62);(32,35,38,37)(33,40,43,42)(34,45,48,47)(36,49,52,53)(39,55,56,57)(41,58,50,60)(44,61,59,62)(46,54,51,63)713 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12712, C31×#12713, C31×#12721, … |
| maximal normal subgroup of | C31×#12712, C31×#12713, C31×#12721, … |
| maximal quotient of | C31×#11434, C31×#11450, C31×#11481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 867);