Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#392109 (order 3072) · Inn(G) = C2×#67 · Out(G) = #379 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | C3×#125 ⋊ C2 |
| Sylow | C3 ⋊ #488 |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,6,7,11)(5,9,8,10)(13,18)(15,23)(17,24)(19,20)(21,27)(25,26);(4,5,7,8)(6,10,11,9)(12,14,16,22)(13,17,20,26)(15,21,23,27)(18,24,19,25);(2,3)(5,8)(9,10)(12,13)(14,17)(15,18)(16,20)(19,23)(21,24)(22,26)(25,27)Polycyclic presentation:
2^6,3|b^2=ef,c^2=e,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,c^b=ce,d^c=df36 conjugacy classes — view the full character table.
| maximal subgroup of | #81150, #81151, #81186, … |
| maximal normal subgroup of | #81150, #81151, #81186, … |
| maximal quotient of | #69547, #69551, #69557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 627);