Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = D3×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×#103 ⋊ C2 |
| Sylow | C3 ⋊ #483 |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,9)(8,11)(12,13,15,17)(14,19,18,16);(4,6,7,10)(5,8,9,11)(13,16)(14,18)(17,19);(2,3)(4,5)(6,8)(7,9)(10,11)Polycyclic presentation:
2^6,3|b^2=d,c^2=f,e^2=f,c^a=cd,g^a=g^2,c^b=ce,e^b=ef,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4805, #78804, #78805, … |
| maximal normal subgroup of | C2×#4805, #78804, #78805, … |
| maximal quotient of | #68789, #68810, #68818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 364);