Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 67 points:
(1,2,3)(4,5,10,15)(6,11,17,33)(7,20,21,55)(8,24,29,60)(9,30,27,14)(12,36,37,22)(13,39,42,28)(16,44,45,54)(18,49,25,59)(19,52,48,35)(23,41,31,56)(26,38,58,43)(32,57,63,46)(34,62,40,51)(47,65,53,67)(50,64,61,66);(4,6)(5,11)(7,16)(8,25)(9,19)(10,17)(12,32)(13,40)(14,35)(15,33)(18,29)(20,54)(21,45)(22,57)(23,47)(24,59)(26,61)(27,48)(28,62)(30,52)(31,53)(34,42)(36,46)(37,63)(38,64)(39,51)(41,67)(43,66)(44,55)(49,60)(50,58)(56,65);(4,7,9,31,10,21,27,23)(5,12,14,43,15,37,30,38)(6,16,19,53,17,45,48,47)(8,26,28,36,29,58,39,22)(11,32,35,66,33,63,52,64)(13,41,24,55,42,56,60,20)(18,50,51,57,25,61,62,46)(34,65,49,54,40,67,59,44);(6,17)(11,33)(16,45)(18,25)(19,48)(32,63)(34,40)(35,52)(44,54)(46,57)(47,53)(49,59)(50,61)(51,62)(64,66)(65,67)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26646);