Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = C2×D3×D4 · Out(G) = C23 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×AGL(1,ℤ/8) ⋊ C2 |
| Sylow | C3 ⋊ #576 |
268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,6)(5,10)(7,15)(8,12)(9,16)(11,19)(13,14)(17,18);(6,13)(7,14)(10,18)(11,17)(12,15)(16,19);(5,9)(6,12)(11,17)(13,15);(2,3)(4,5)(6,10)(7,11)(8,9)(12,16)(13,18)(14,17)(15,19)Polycyclic presentation:
2^6,3|e^2=f,b^a=bf,g^a=g^2,d^b=df,d^c=de,e^c=ef,e^d=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | #80280, #80282, #80287, … |
| maximal normal subgroup of | #80280, #80282, #80287, … |
| maximal quotient of | #78868, #78872, #78874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1333);