Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#12026 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #427 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| socle | C2×C3 | #1568 |
| prime radical | C23×C3 | #108 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C3×#97 ◁ C3×#405 ◁ G | |
180 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2,3)(4,5,9,16,11,17,34,46,12,18,33,15,10,37,8,32)(6,19,22,39,24,48,21,40,25,44,35,61,23,57,55,13)(7,26,28,59,30,43,27,45,31,60,36,62,29,14,41,42)(20,49,51,38,53,63,65,58,54,67,56,64,52,47,50,66);(4,6,8,21,10,23,33,22,12,25,34,55,11,24,9,35)(5,13,15,19,17,39,32,48,18,40,16,44,37,61,46,57)(7,20,27,50,29,52,28,56,31,54,41,65,30,53,36,51)(14,38,42,47,26,58,59,49,43,64,45,63,60,66,62,67);(4,7,10,29,12,31,11,30)(5,14,17,26,18,43,37,60)(6,20,23,52,25,54,24,53)(8,27,33,28,34,41,9,36)(13,38,39,58,40,64,61,66)(15,42,32,59,16,45,46,62)(19,47,48,49,44,63,57,67)(21,50,22,56,55,65,35,51)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5014);