Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#2196 (order 1536) · Inn(G) = #937 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C4⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #456 |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,7,9,6)(5,11,13,19)(8,14,18,16)(10,17,15,12);(4,6,9,7)(5,10,13,15)(8,16,18,14)(11,12,19,17);(2,3)(4,5)(6,10)(7,15)(8,17)(9,13)(11,16)(12,18)(14,19)Polycyclic presentation:
2^6,3|b^2=f,c^2=f,d^2=f,c^a=cd,d^a=df,e^a=ef,g^a=g^2,c^b=ce,d^b=df,d^c=df30 conjugacy classes — view the full character table.
| maximal subgroup of | #67463, #67550, #69167, … |
| maximal normal subgroup of | #67463, #67550, #69167, … |
| maximal quotient of | #69573, #69592, #69593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 654);