Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#2196 (order 1536) · Inn(G) = #953 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #456 |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,7,9,5)(6,13,14,10)(8,11,18,16)(12,17,19,15);(4,6)(5,10)(7,13)(8,17)(9,14)(11,19)(12,16)(15,18);(2,3)(4,5,9,7)(6,12,14,19)(8,16,18,11)(10,17,13,15)Polycyclic presentation:
2^6,3|a^2=f,c^2=f,e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^2,d^b=df,e^b=ef,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | #70988, #70989, #70993, … |
| maximal normal subgroup of | #70988, #70989, #70993, … |
| maximal quotient of | #71658, #71660, #71664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 805);