Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13406 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C2×C3 | #2034 |
| prime radical | C3×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
390 conjugacy classes of subgroups.
| 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)(6,17)(7,23)(8,15)(9,31)(10,16)(11,19)(12,26)(13,30)(14,29)(18,49)(20,36)(21,38)(22,37)(24,57)(25,58)(27,44)(28,56)(32,43)(33,42)(34,45)(35,54)(39,65)(40,61)(41,62)(46,60)(47,51)(48,55)(50,67)(52,66)(53,64)(59,63);(4,6)(5,11)(7,24)(8,21)(9,32)(10,22)(12,39)(13,37)(14,43)(15,17)(16,38)(18,27)(19,30)(20,34)(23,49)(25,52)(26,59)(28,50)(29,53)(31,48)(33,51)(35,40)(36,46)(41,47)(42,63)(44,67)(45,64)(54,62)(55,60)(56,66)(57,58)(61,65);(4,7,10,28)(5,12,16,41)(6,18)(8,27,30,25)(9,33,34,61)(11,35)(13,42,15,40)(14,44,46,58)(17,47)(19,24)(20,54)(21,50)(22,52)(23,45,56,31)(26,60,62,29)(32,59)(36,49)(37,39)(38,63)(43,66)(48,67)(51,55)(53,65)(57,64)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6394);