Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23893 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
1610 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)(6,17)(7,23)(8,29)(9,33)(10,15)(11,16)(12,20)(13,27)(14,30)(18,46)(19,35)(21,37)(22,36)(24,44)(25,55)(26,41)(28,40)(31,43)(32,42)(34,48)(38,52)(39,58)(45,49)(47,64)(50,61)(51,60)(53,56)(54,59)(57,63)(62,67)(65,66);(4,6)(5,12)(7,24)(8,19)(9,21)(10,20)(11,22)(13,38)(14,35)(15,36)(16,37)(17,33)(18,26)(23,34)(25,54)(27,48)(28,52)(29,45)(30,51)(31,49)(32,50)(39,62)(40,46)(41,44)(42,60)(43,61)(47,57)(53,59)(55,64)(56,67)(58,66)(63,65);(4,7)(5,13)(6,18)(8,25)(9,26)(10,27)(11,28)(12,34)(14,39)(15,40)(16,41)(17,44)(19,47)(20,24)(21,52)(22,48)(23,33)(29,53)(30,56)(31,57)(32,58)(35,59)(36,38)(37,46)(42,63)(43,55)(45,64)(49,66)(50,67)(51,54)(60,62)(61,65);(4,8,11,32)(5,14,16,43)(6,19,22,50)(7,25,28,58)(9,31,10,30)(12,35,37,61)(13,39,41,55)(15,42,33,29)(17,45,36,60)(18,47,48,67)(20,51,21,49)(23,53,40,63)(24,54,52,66)(26,57,27,56)(34,59,46,65)(38,62,44,64)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16881);