Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2033 |
| 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 | |
382 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)(7,22,26,57)(8,28,30,13)(9,31,33,62)(11,20,37,18)(12,38,41,25)(14,43,45,29)(17,48,23,59)(19,35,54,66)(24,58,27,44)(32,42,61,40)(34,52,39,64)(46,60,67,50)(47,51,63,53)(49,65,55,56);(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)(22,56)(24,49)(25,60)(27,55)(29,53)(31,63)(32,64)(33,54)(34,41)(38,50)(40,46)(42,67)(43,51)(44,48)(45,66)(47,62)(52,61)(57,65)(58,59);(4,7,8,24,10,26,30,27)(5,12,13,40,15,41,28,42)(6,17,18,49,21,23,20,55)(9,32,29,38,33,61,43,25)(11,34,16,46,37,39,36,67)(14,44,31,57,45,58,62,22)(19,52,51,50,54,64,53,60)(35,59,47,65,66,48,63,56)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6378);