Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#72214076 (order 6144) · Inn(G) = D3×D4 · Out(G) = #2403 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×C3×Q16 ⋊ C2 |
| Sylow | C3 ⋊ #495 |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(5,7)(8,10,12,19)(9,15,16,23)(11,20,14,18)(13,17,21,22);(8,9,14,21,12,16,11,13)(10,17,20,23,19,22,18,15);(2,3)(4,5)(6,7)(9,13)(11,14)(15,17)(16,21)(18,20)(22,23)Polycyclic presentation:
2^6,3|b^2=ef,c^2=f,e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^2,c^b=cef,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69216, #69217, #69235, … |
| maximal normal subgroup of | #69216, #69217, #69235, … |
| maximal quotient of | #70226, #70251, #70257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 750);