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×#12027 |
| 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)(4,5,9,16,11,17,35,48,12,18,34,15,10,38,8,33)(6,19,22,40,24,50,21,41,25,46,36,64,23,58,56,13)(7,26,29,60,31,45,28,47,32,61,37,65,30,14,42,44)(20,51,53,39,55,66,63,67,27,62,57,43,54,49,52,59);(4,6,8,21,10,23,34,22,12,25,35,56,11,24,9,36)(5,13,15,19,17,40,33,50,18,41,16,46,38,64,48,58)(7,27,28,63,30,55,29,53,32,20,42,52,31,54,37,57)(14,43,44,66,26,59,60,62,45,39,47,49,61,67,65,51);(4,7,10,30,12,32,11,31)(5,14,17,26,18,45,38,61)(6,20,23,54,25,27,24,55)(8,28,34,29,35,42,9,37)(13,39,40,67,41,43,64,59)(15,44,33,60,16,47,48,65)(19,49,50,51,46,66,58,62)(21,52,22,57,56,63,36,53)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5015);