Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#61451 (order 1536) · Inn(G) = C2×#213 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×D8 ⋊ C2 |
| Sylow | C3 ⋊ #509 |
90 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)(5,11)(6,8)(7,15)(10,12)(13,19)(14,16)(17,18);(4,6,9,8)(5,10,13,17)(7,14,16,15)(11,18,19,12);(2,3)(4,5)(6,10)(7,11)(8,17)(9,13)(12,15)(14,18)(16,19)Polycyclic presentation:
2^6,3|b^2=f,d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^2,c^b=cf,d^c=de,e^c=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #69254, #69262, #69263, … |
| maximal normal subgroup of | #69254, #69262, #69263, … |
| maximal quotient of | #70596, #70598, #70600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 706);