Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #884 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#101 ⋊ C2 |
| Sylow | C3 ⋊ #473 |
100 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,7,9,20)(5,11,13,26)(6,14,16,18)(8,21,22,19)(10,23,25,34)(12,27,28,17)(15,31,32,30)(24,29,35,33);(4,6,8,15,9,16,22,32)(5,10,12,24,13,25,28,35)(7,18,19,31,20,14,21,30)(11,23,17,33,26,34,27,29);(2,3)(4,5)(6,10)(7,17)(8,12)(9,13)(11,21)(14,29)(15,24)(16,25)(18,33)(19,26)(20,27)(22,28)(23,30)(31,34)(32,35)Polycyclic presentation:
2^6,3|b^2=d,c^2=f,d^2=f,c^a=cde,e^a=ef,g^a=g^2,c^b=cef,e^b=ef,d^c=df27 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4755, #78712, #78719, … |
| maximal normal subgroup of | C2×#4755, #78712, #78719, … |
| maximal quotient of | #68441, #68443, #68444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 314);