Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2053 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C31 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C31 |
| socle | C2×C31 | #394 |
| prime radical | C31×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C31 ◁ C22×C4×C31 ◁ G | |
264 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 80 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)(34,43)(35,48)(36,53)(37,41)(38,42)(39,45)(40,50)(44,54)(46,56)(47,55)(49,57)(51,58)(52,61)(59,62)(60,63);(32,34)(33,39)(35,49)(36,46)(37,45)(38,47)(40,57)(41,55)(42,56)(43,53)(44,51)(48,54)(50,60)(52,59)(58,62)(61,63);(32,35,36,50,38,52,37,51)(33,40,41,48,42,58,53,61)(34,44,45,59,47,60,46,49)(39,54,43,57,56,63,55,62)806 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12170, C31×#12171, C31×#12172, … |
| maximal normal subgroup of | C31×#12170, C31×#12171, C31×#12172, … |
| maximal quotient of | C31×#10464, C31×#10465, C31×#10466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 785);