Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13402 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
384 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)(4,5,10,15)(6,16,21,36)(7,22,27,57)(8,28,30,13)(9,31,33,55)(11,20,37,18)(12,38,42,25)(14,43,45,29)(17,47,50,67)(19,35,54,65)(23,58,48,66)(24,56,26,44)(32,41,59,40)(34,62,64,52)(39,61,63,49)(46,51,60,53);(4,6)(5,11)(7,23)(8,20)(9,19)(10,21)(12,39)(13,36)(14,35)(15,37)(16,28)(17,26)(18,30)(22,47)(24,50)(25,52)(27,48)(29,53)(31,60)(32,61)(33,54)(34,41)(38,62)(40,64)(42,63)(43,51)(44,66)(45,65)(46,55)(49,59)(56,58)(57,67);(4,7,10,27)(5,12,15,42)(6,17)(8,26,30,24)(9,32,33,59)(11,34)(13,41,28,40)(14,44,45,56)(16,39)(18,23)(19,52)(20,48)(21,50)(22,55,57,31)(25,43,38,29)(35,47)(36,63)(37,64)(46,66)(49,53)(51,61)(54,62)(58,60)(65,67)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6391);