Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#392113 (order 3072) · Inn(G) = C2×#67 · Out(G) = #379 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #499 |
86 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)(4,6,7,11)(5,8,9,10)(13,18)(14,21)(15,23)(17,24)(19,20)(25,26);(4,5,7,9)(6,10,11,8)(12,14)(13,17)(15,22)(16,21)(18,26)(19,24)(20,25)(23,27);(2,3)(12,13,15,19,16,20,23,18)(14,17,22,24,21,25,27,26)Polycyclic presentation:
2^6,3|a^2=e,b^2=d,c^2=d,e^2=f,c^a=ce,g^a=g^2,c^b=cdf,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #81156, #81157, #81160, … |
| maximal normal subgroup of | #81156, #81157, #81160, … |
| maximal quotient of | #69541, #69549, #69554, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 624);