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 ⋊ #367 |
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)(4,5,6,7)(9,13)(10,17)(11,18)(14,21)(15,22)(16,20)(19,23);(2,3)(8,9,10,14,18,21,23,13,12,16,19,22,11,15,17,20)Polycyclic presentation:
2^6,3|a^2=d,b^2=c,d^2=ef,e^2=f,b^a=bdf,g^a=g^2,d^b=de,e^b=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69175, #69177, #69179, … |
| maximal normal subgroup of | #69175, #69177, #69179, … |
| maximal quotient of | #67470, #67579, #67582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 49);