Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#2196 (order 1536) · Inn(G) = D3×C22⋊C4 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#101 ⋊ C2 |
| Sylow | C3 ⋊ #413 |
150 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)(4,6,7,13,8,14,16,19)(5,10,11,17,9,12,15,18);(4,5,8,9)(6,12,14,10)(7,15,16,11)(13,17,19,18);(2,3)(5,9)(10,12)(11,15)(17,18)Polycyclic presentation:
2^6,3|b^2=f,c^2=d,d^2=f,b^a=bf,g^a=g^2,c^b=ce,d^b=df,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | #69165, #69166, #69172, … |
| maximal normal subgroup of | #69165, #69166, #69172, … |
| maximal quotient of | #68124, #68126, #68129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 310);