Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = D3×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #479 |
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)(4,7,8,6)(5,9,10,11)(14,20)(15,23)(17,24)(18,27)(21,22)(25,26);(4,6,8,7)(5,9,10,11)(12,14)(13,17)(15,20)(16,22)(18,24)(19,26)(21,23)(25,27);(2,3)(4,5,8,10)(6,9,7,11)(12,13,16,19)(14,17,22,26)(15,18,23,27)(20,24,21,25)Polycyclic presentation:
2^6,3|a^2=e,b^2=ef,c^2=ef,d^2=f,c^a=cef,g^a=g^2,c^b=cd,d^b=df,d^c=df33 conjugacy classes — view the full character table.
| maximal subgroup of | #79155, #79159, #79162, … |
| maximal normal subgroup of | #79155, #79159, #79162, … |
| maximal quotient of | #68789, #68802, #68809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 436);