Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#33385 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C2×C3 | C22×#102 |
| prime radical | C3×#3584 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
1114 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)(4,5,9,15,11,16,10,31)(6,17,20,43,21,44,32,40)(7,13,24,39,22,37,25,54)(8,26,28,41,30,56,29,14)(12,27,34,48,36,57,35,19)(18,42,47,64,45,63,55,62)(23,50,52,61,49,66,53,38)(33,51,59,65,58,67,60,46);(4,6)(5,12)(7,18)(8,27)(9,20)(10,32)(11,21)(13,33)(14,40)(15,34)(16,36)(17,26)(19,29)(22,45)(23,51)(24,47)(25,55)(28,48)(30,57)(31,35)(37,58)(38,62)(39,59)(41,43)(42,50)(44,56)(46,53)(49,67)(52,65)(54,60)(61,64)(63,66);(4,7)(5,13)(6,18)(8,23)(9,24)(10,25)(11,22)(12,33)(14,38)(15,39)(16,37)(17,42)(19,46)(20,47)(21,45)(26,50)(27,51)(28,52)(29,53)(30,49)(31,54)(32,55)(34,59)(35,60)(36,58)(40,62)(41,61)(43,64)(44,63)(48,65)(56,66)(57,67);(7,22)(13,37)(18,45)(23,49)(24,25)(33,58)(38,61)(39,54)(42,63)(46,65)(47,55)(50,66)(51,67)(52,53)(59,60)(62,64)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26373);