Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13392 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C2×C3 | #2033 |
| prime radical | C3×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ 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)(6,16)(7,22)(8,28)(9,31)(10,15)(11,18)(12,25)(13,30)(14,29)(17,47)(19,36)(20,37)(21,38)(23,57)(24,55)(26,56)(27,43)(32,42)(33,40)(34,61)(35,63)(39,60)(41,59)(44,62)(45,49)(46,52)(48,64)(50,66)(51,65)(53,54)(58,67);(4,6)(5,11)(7,23)(8,20)(9,32)(10,21)(12,35)(13,38)(14,42)(15,37)(16,28)(17,26)(18,30)(19,34)(22,54)(24,48)(25,49)(27,53)(29,50)(31,46)(33,51)(36,44)(39,67)(40,45)(41,65)(43,47)(52,62)(55,57)(56,64)(58,59)(60,63)(61,66);(4,7,8,24,10,26,30,27)(5,12,13,39,15,41,28,40)(6,17,18,48,21,23,20,53)(9,33,29,59,34,60,62,25)(11,35,16,45,37,65,38,67)(14,43,31,56,44,55,61,22)(19,51,50,49,32,63,52,58)(36,57,46,64,42,47,66,54)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6380);