Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33540 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
1078 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)(6,18)(7,23)(8,27)(9,17)(10,16)(11,14)(12,20)(13,25)(15,30)(19,46)(21,52)(22,45)(24,55)(26,42)(28,43)(29,39)(31,41)(32,37)(33,40)(34,61)(35,51)(36,59)(38,58)(44,48)(47,53)(49,66)(50,64)(54,57)(56,62)(60,65)(63,67);(4,6,11,22)(5,12,14,35)(7,19,26,50)(8,28,31,59)(9,21,10,33)(13,34,37,62)(15,40,39,52)(16,36,17,43)(18,41,45,27)(20,29,51,30)(23,53,42,63)(24,56,58,61)(25,48,32,49)(38,64,55,46)(44,65,66,54)(47,60,67,57);(4,7)(5,13)(6,19)(8,24)(9,32)(10,25)(11,26)(12,34)(14,37)(15,38)(16,42)(17,23)(18,44)(20,47)(21,49)(22,50)(27,54)(28,56)(29,60)(30,57)(31,58)(33,48)(35,62)(36,63)(39,55)(40,64)(41,65)(43,53)(45,66)(46,52)(51,67)(59,61);(5,14)(12,35)(13,37)(15,39)(16,17)(18,45)(23,42)(27,41)(34,62)(36,43)(38,55)(40,52)(44,66)(46,64)(53,63)(54,65)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26528);