Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of D48 |
| automorphism | Aut(G) = D3×#495725 (order 3072) · Inn(G) = D24 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #366 |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(8,9,14,20,11,15,18,21,12,16,22,23,17,19,10,13);(2,3)(4,5,6,7)(9,13)(10,14)(11,17)(15,23)(16,21)(18,22)(19,20)Polycyclic presentation:
2^6,3|a^2=c,b^2=de,d^2=ef,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^254 conjugacy classes — view the full character table.
| maximal subgroup of | #67575, #67576, #67591, … |
| maximal normal subgroup of | #67575, #67576, #67591, … |
| maximal quotient of | #67500, #67591, #67592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 63);