Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13310 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13311 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C3×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
520 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)(6,18)(7,24)(8,28)(9,17)(10,15)(11,16)(12,20)(13,26)(14,31)(19,48)(21,41)(22,46)(23,47)(25,57)(27,44)(29,35)(30,43)(32,42)(33,40)(34,58)(36,53)(37,52)(38,63)(39,60)(45,50)(49,55)(51,65)(54,64)(56,59)(61,66)(62,67);(4,6,11,23)(5,12,16,36)(7,19,27,51)(8,29,32,52)(9,22,10,21)(13,38,40,34)(14,41,43,46)(15,37,17,35)(18,42,47,28)(20,30,53,31)(24,55,44,62)(25,58,60,63)(26,54,33,50)(39,65,57,48)(45,56,64,66)(49,61,67,59);(4,7)(5,13)(6,19)(8,25)(9,33)(10,26)(11,27)(12,34)(14,39)(15,44)(16,40)(17,24)(18,45)(20,49)(21,54)(22,50)(23,51)(28,56)(29,58)(30,61)(31,59)(32,60)(35,62)(36,38)(37,55)(41,48)(42,66)(43,57)(46,65)(47,64)(52,63)(53,67)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6299);