Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×D4×AGL(1,ℤ/8) (order 1536) · Inn(G) = #930 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #473 |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,7)(5,12)(6,17)(8,24)(9,22)(10,23)(11,25)(13,29)(14,21)(15,28)(16,30)(18,34)(19,32)(20,33)(26,35)(27,31);(4,6,8,18,10,20,9,19)(5,11,13,26,15,27,14,16)(7,17,24,34,23,33,22,32)(12,25,29,35,28,31,21,30);(2,3)(4,5,10,15)(6,16,20,26)(7,21,23,29)(8,14,9,13)(11,18,27,19)(12,24,28,22)(17,31,33,25)(30,34,35,32)Polycyclic presentation:
2^6,3|a^2=f,b^2=ef,d^2=f,e^2=f,b^a=be,c^a=cd,d^a=df,e^a=ef,g^a=g^2,d^c=df36 conjugacy classes — view the full character table.
| maximal subgroup of | #69128, #69129, #71093, … |
| maximal normal subgroup of | #69128, #69129, #71093, … |
| maximal quotient of | #69891, #69894, #69904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 702);