Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13397 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
390 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)(4,5,11,17)(6,12,21,37)(7,22,24,56)(8,28,32,44)(9,16,10,15)(13,38,40,25)(14,30,45,31)(18,47,48,67)(19,50,53,66)(20,46,33,36)(23,58,54,55)(26,57,27,43)(29,42,60,41)(34,62,63,49)(35,61,64,52)(39,59,65,51);(4,6)(5,12)(7,23)(8,19)(9,33)(10,20)(11,21)(13,39)(14,35)(15,46)(16,36)(17,37)(18,26)(22,55)(24,54)(25,59)(27,48)(28,50)(29,49)(30,61)(31,52)(32,53)(34,41)(38,51)(40,65)(42,63)(43,47)(44,66)(45,64)(56,58)(57,67)(60,62);(4,7,10,26,11,24,9,27)(5,13,16,41,17,40,15,42)(6,18,20,54,21,48,33,23)(8,29,31,38,32,60,30,25)(12,34,36,65,37,63,46,39)(14,43,28,56,45,57,44,22)(19,51,52,62,53,59,61,49)(35,58,50,67,64,55,66,47)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6385);