Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#21070 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C8 ◁ G | |
1026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(5,6)(8,9,13,18,14,17,12,26)(10,15,21,30,22,29,20,36)(11,23,24,16,25,32,27,31)(19,33,34,28,35,39,37,38);(4,5,7,6)(8,10,13,21,14,22,12,20)(9,15,18,30,17,29,26,36)(11,19,24,34,25,35,27,37)(16,28,32,39,31,38,23,33);(4,6,7,5)(8,11)(9,16)(10,19)(12,27)(13,24)(14,25)(15,28)(17,31)(18,32)(20,37)(21,34)(22,35)(23,26)(29,38)(30,39)(33,36);(9,17)(15,29)(16,31)(18,26)(23,32)(28,38)(30,36)(33,39)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14058);