Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×#120 (order 3072) · Inn(G) = C2×D12 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #462 |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(5,7)(8,10,17,21,12,19,11,18)(9,13,20,23,15,22,14,16);(4,5,6,7);(2,3)(8,9,12,15)(10,16,19,23)(11,20,17,14)(13,21,22,18)Polycyclic presentation:
2^6,3|a^2=f,b^2=e,c^2=df,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce39 conjugacy classes — view the full character table.
| maximal subgroup of | #68190, #68191, #68192, … |
| maximal normal subgroup of | #68190, #68191, #68192, … |
| maximal quotient of | #68547, #68552, #68553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 408);