Finite Group DatabasebetaFinite Group Databasebeta| order | 3968 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C31 · Schur multiplier (H2) = C23 |
| direct product | C31×#2123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C31 | C2×#99 |
| commutator | C23 | C22×C4×C31 |
| Frattini, cosocle | C2×D4 | C23×C31 |
| prime radical | C2×#41×C31 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×#41×C31 ◁ G | |
234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 234 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)(34,42)(35,39)(36,49,50,40)(37,41)(38,44,52,59)(43,56)(45,57)(46,60)(47,62,63,54)(48,55)(51,58,53,61);(32,34,37,45)(33,38,41,52)(35,46,48,43)(36,44,50,59)(39,53,55,51)(40,42,49,57)(47,61,63,58)(54,60,62,56);(32,35,37,48)(33,39,41,55)(34,43,45,46)(36,47,50,63)(38,51,52,53)(40,54,49,62)(42,56,57,60)(44,58,59,61)713 conjugacy classes — view the full character table.
| maximal subgroup of | C31×#12716, C31×#12717, C31×#12723, … |
| maximal normal subgroup of | C31×#12716, C31×#12717, C31×#12723, … |
| maximal quotient of | C31×#11428, C31×#11454, C31×#11474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3968, 855);