Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#442 (order 768) · Inn(G) = D3×D8 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×SD32 ⋊ C2 |
| Sylow | C3 ⋊ #510 |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(5,11)(6,14)(7,18)(8,19)(10,22)(12,29)(15,16)(17,33)(20,34)(21,32)(23,31)(24,35)(25,28)(26,30);(4,6,9,17)(5,10,11,25)(7,14,20,33)(8,21,19,16)(12,28,26,22)(13,31,27,24)(15,34,32,18)(23,29,35,30);(2,3)(4,5)(6,10)(7,12)(8,13)(9,11)(14,28)(15,23)(16,24)(17,25)(18,30)(19,27)(20,26)(21,31)(22,33)(29,34)(32,35)Polycyclic presentation:
2^6,3|b^2=f,d^2=ef,e^2=f,c^a=cf,g^a=g^2,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef27 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4915, #82024, #82030, … |
| maximal normal subgroup of | C2×#4915, #82024, #82030, … |
| maximal quotient of | #69176, #69178, #69185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 474);