Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #939 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #473 |
100 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 37 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)(4,7,10,24)(5,12,15,30)(6,17,19,33)(8,23,9,22)(11,26,27,35)(13,29,14,20)(16,32,31,28)(18,34,25,21);(4,6)(5,11)(7,21)(8,18)(9,25)(10,19)(12,28)(13,16)(14,31)(15,27)(17,23)(20,35)(22,33)(24,34)(26,29)(30,32);(2,3)(4,5,10,15)(6,16,19,31)(7,20,24,29)(8,14,9,13)(11,25,27,18)(12,23,30,22)(17,26,33,35)(21,32,34,28)Polycyclic presentation:
2^6,3|a^2=f,c^2=f,d^2=f,e^2=f,b^a=be,c^a=cd,d^a=df,e^a=ef,g^a=g^2,c^b=ce,e^b=ef,d^c=df,e^c=ef27 conjugacy classes — view the full character table.
| maximal subgroup of | #80295, #80296, #80301, … |
| maximal normal subgroup of | #80295, #80296, #80301, … |
| maximal quotient of | #70300, #70301, #70304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 763);