Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33538 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C2×C3 | C2×#447 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
1088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 67 points:
(1,2,3)(4,5)(6,18)(7,23)(8,14)(9,29)(10,16)(11,17)(12,21)(13,26)(15,31)(19,47)(20,45)(22,46)(24,39)(25,56)(27,43)(28,42)(30,35)(32,41)(33,40)(34,61)(36,60)(37,52)(38,59)(44,50)(48,66)(49,53)(51,64)(54,58)(55,65)(57,63)(62,67);(4,6)(5,12)(7,19)(8,20)(9,30)(10,33)(11,22)(13,34)(14,35)(15,40)(16,36)(17,37)(18,29)(21,31)(23,53)(24,48)(25,57)(26,50)(27,51)(28,52)(32,60)(38,64)(39,63)(41,46)(42,45)(43,62)(44,54)(47,56)(49,58)(55,67)(59,61)(65,66);(4,7)(5,13)(6,19)(8,24)(9,25)(10,26)(11,27)(12,34)(14,23)(15,38)(16,43)(17,39)(18,44)(20,48)(21,49)(22,51)(28,55)(29,54)(30,57)(31,58)(32,59)(33,50)(35,53)(36,62)(37,63)(40,64)(41,65)(42,56)(45,47)(46,66)(52,67)(60,61);(4,8,11,10)(5,14,17,16)(6,20,22,33)(7,24,27,26)(9,28,32,31)(12,35,37,36)(13,23,39,43)(15,29,42,41)(18,45,46,40)(19,48,51,50)(21,30,52,60)(25,55,59,58)(34,53,63,62)(38,54,56,65)(44,47,66,64)(49,57,67,61)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26526);