Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#72213256 (order 6144) · Inn(G) = C2×D12 · 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)(8,10)(12,14,22,26,16,21,15,23)(13,18,24,27,17,25,19,20);(4,6,7,11)(5,8,9,10)(13,17)(14,21)(19,24)(23,26);(2,3)(4,5,7,9)(6,10,11,8)(12,13,16,17)(14,20,21,27)(15,24,22,19)(18,26,25,23)Polycyclic presentation:
2^6,3|a^2=e,b^2=ef,c^2=de,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,c^b=cf36 conjugacy classes — view the full character table.
| maximal subgroup of | #67521, #68358, #68359, … |
| maximal normal subgroup of | #67521, #68358, #68359, … |
| maximal quotient of | #68631, #68638, #68642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 262);