Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #826 |
| automorphism | Aut(G) = C22×D3×#442 (order 1536) · Inn(G) = #213 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #358 |
58 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)(4,5,6,7)(9,13)(10,17)(11,18)(14,15)(16,22)(19,23)(20,21);(2,3)(8,9,12,16)(10,13,19,22)(11,20,18,15)(14,23,21,17)Polycyclic presentation:
2^6,3|a^2=f,b^2=c,d^2=ef,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^2,d^b=de,e^b=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69184, #69185, #69188, … |
| maximal normal subgroup of | #69184, #69185, #69188, … |
| maximal quotient of | #67472, #67474, #67476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 51);