Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = D3×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #441 |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,12,18)(9,13,15,16)(11,20,19,17)(14,23,22,21);(4,6,7,5)(10,17)(11,19)(13,21)(14,22)(16,23)(18,20);(2,3)(4,5,7,6)(8,9,12,15)(10,16,18,13)(11,14,19,22)(17,23,20,21)Polycyclic presentation:
2^6,3|a^2=e,b^2=ef,c^2=f,d^2=f,c^a=cf,g^a=g^2,c^b=cd,d^b=df,d^c=df36 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4789, #78774, #78810, … |
| maximal normal subgroup of | C2×#4789, #78774, #78810, … |
| maximal quotient of | #68215, #68221, #68222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 348);