Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C23×D3×#120 (order 1536) · Inn(G) = C22×D3 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×C4⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #431 |
118 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)(4,5,6,8,7,9,10,11)(13,17)(14,18);(12,14,15,18)(13,16,17,19);(2,3)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^6,3|b^2=e,c^2=d,d^2=f,c^a=ce,g^a=g^2,c^b=ce60 conjugacy classes — view the full character table.
| maximal subgroup of | #68165, #68167, #68169, … |
| maximal normal subgroup of | #68165, #68167, #68169, … |
| maximal quotient of | #68596, #68599, #68607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 392);