Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#72215520 (order 6144) · Inn(G) = C2×D12 · Out(G) = C22×#120 |
| 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)(12,14,15,21,16,22,24,26)(13,18,19,20,17,25,23,27);(4,6,7,11)(5,8,9,10)(13,17)(18,25)(19,23)(20,27);(2,3)(4,5,7,9)(6,10,11,8)(12,13,16,17)(14,20,22,27)(15,23,24,19)(18,26,25,21)Polycyclic presentation:
2^6,3|a^2=e,b^2=ef,c^2=d,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^254 conjugacy classes — view the full character table.
| maximal subgroup of | #67468, #67563, #67573, … |
| maximal normal subgroup of | #67468, #67563, #67573, … |
| maximal quotient of | #68634, #68641, #69766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 242);