Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×B(4,𝔽2) (order 768) · Inn(G) = #834 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #352 |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,6,8,14)(5,9,11,17)(7,13,16,19)(10,12,15,18);(2,3)(4,5)(6,12)(7,15,16,10)(8,11)(9,13,17,19)(14,18)Polycyclic presentation:
2^6,3|a^2=c,b^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^2,c^b=ce,d^c=df27 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#352, #69143, #69144, … |
| maximal normal subgroup of | D3×#352, #69143, #69144, … |
| maximal quotient of | #67627, #67636, #67637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 98);