Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#72213256 (order 6144) · Inn(G) = C2×#67 · Out(G) = #2021 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #491 |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(6,10)(12,14,16,22)(13,18,17,25)(15,23,24,21)(19,27,26,20);(4,6,7,10)(5,8,9,11)(13,17)(18,25)(19,26)(20,27);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13,15,19,16,17,24,26)(14,20,21,25,22,27,23,18)Polycyclic presentation:
2^6,3|a^2=e,b^2=d,c^2=f,e^2=f,b^a=bf,c^a=cde,g^a=g^2,c^b=cd,e^c=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #67551, #71246, #71247, … |
| maximal normal subgroup of | #67551, #71246, #71247, … |
| maximal quotient of | #69571, #69575, #69583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 641);