Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #826 |
| automorphism | Aut(G) = C2×D3×#442 (order 768) · Inn(G) = #213 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×#108 ⋊ C2 |
| Sylow | C3 ⋊ #359 |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,6,7,16,10,18,9,25)(5,11,12,26,14,28,24,27)(8,21,19,33,23,35,22,17)(13,29,20,15,30,31,34,32);(2,3)(4,5)(6,15)(7,12)(8,20)(9,24)(10,14)(11,17)(13,22)(16,31)(18,32)(19,30)(21,26)(23,34)(25,29)(27,35)(28,33)Polycyclic presentation:
2^6,3|b^2=c,c^2=f,d^2=ef,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^2,d^b=de,e^b=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#359, #69241, #69258, … |
| maximal normal subgroup of | D3×#359, #69241, #69258, … |
| maximal quotient of | #67467, #67468, #67471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 56);