Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13405 |
| 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 | |
386 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)(6,16,21,37)(7,22,26,59)(8,27,30,13)(9,31,34,58)(11,20,38,18)(12,39,42,25)(14,44,47,29)(17,50,23,60)(19,36,32,45)(24,62,28,46)(33,43,63,41)(35,64,40,55)(48,61,65,52)(49,54,66,56)(51,67,53,57);(4,6)(5,11)(7,23)(8,20)(9,32)(10,21)(12,40)(13,37)(14,45)(15,38)(16,27)(17,26)(18,30)(19,34)(22,57)(24,51)(25,52)(28,53)(29,54)(31,49)(33,55)(35,42)(36,47)(39,61)(41,65)(43,48)(44,56)(46,50)(58,66)(59,67)(60,62)(63,64);(4,7,10,26)(5,12,15,42)(6,17)(8,28,30,24)(9,33,34,63)(11,35)(13,43,27,41)(14,46,47,62)(16,48)(18,53)(19,55)(20,51)(21,23)(22,58,59,31)(25,44,39,29)(32,64)(36,50)(37,65)(38,40)(45,60)(49,67)(52,56)(54,61)(57,66)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6393);