Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13401 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #482 |
| 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)(6,16)(7,22)(8,27)(9,31)(10,15)(11,18)(12,25)(13,30)(14,29)(17,45)(19,35)(20,36)(21,37)(23,56)(24,58)(26,55)(28,41)(32,40)(33,54)(34,50)(38,61)(39,60)(42,59)(43,47)(44,51)(46,67)(48,53)(49,62)(52,66)(57,64)(63,65);(4,6)(5,11)(7,23)(8,20)(9,19)(10,21)(12,34)(13,37)(14,35)(15,36)(16,27)(17,26)(18,30)(22,53)(24,46)(25,57)(28,48)(29,51)(31,62)(32,63)(33,52)(38,43)(39,65)(40,64)(41,45)(42,66)(44,54)(47,60)(49,59)(50,61)(55,67)(56,58);(4,7,10,26)(5,12,15,39)(6,17)(8,28,30,24)(9,32,33,61)(11,34)(13,40,27,38)(14,41,42,58)(16,43)(18,48)(19,50)(20,46)(21,23)(22,54,55,31)(25,59,60,29)(35,45)(36,65)(37,64)(44,67)(47,51)(49,57)(52,63)(53,62)(56,66)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6389);