Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
1084 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)(4,5,11,17)(6,18,22,46)(7,23,27,38)(8,16,10,14)(9,29,32,41)(12,34,37,21)(13,24,40,26)(15,28,43,31)(19,47,51,65)(20,52,33,42)(25,56,59,39)(30,44,60,36)(35,57,63,67)(45,48,66,50)(49,53,61,62)(54,55,64,58);(4,6)(5,12)(7,19)(8,20)(9,30)(10,33)(11,22)(13,35)(14,36)(15,42)(16,44)(17,37)(18,29)(21,31)(23,53)(24,48)(25,57)(26,50)(27,51)(28,34)(32,60)(38,62)(39,65)(40,63)(41,46)(43,52)(45,54)(47,56)(49,58)(55,61)(59,67)(64,66);(4,7)(5,13)(6,19)(8,24)(9,25)(10,26)(11,27)(12,35)(14,38)(15,39)(16,23)(17,40)(18,45)(20,48)(21,49)(22,51)(28,55)(29,54)(30,57)(31,58)(32,59)(33,50)(34,61)(36,62)(37,63)(41,64)(42,65)(43,56)(44,53)(46,66)(47,52)(60,67);(4,8,11,10)(5,14,17,16)(6,20,22,33)(7,24,27,26)(9,28,32,31)(12,36,37,44)(13,38,40,23)(15,41,43,29)(18,42,46,52)(19,48,51,50)(21,30,34,60)(25,55,59,58)(35,62,63,53)(39,64,56,54)(45,65,66,47)(49,57,61,67)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26530);