Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#6628842 (order 3072) · Inn(G) = D3×D4 · Out(G) = C2×#120 |
| 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)(4,6,7,5)(9,14)(10,16)(11,19)(13,20)(17,18)(21,23);(4,5,7,6)(8,10,12,18)(9,13,14,21)(11,16,19,17)(15,23,22,20);(2,3)(4,5,7,6)(8,9)(10,13)(11,15)(12,14)(16,23)(17,20)(18,21)(19,22)Polycyclic presentation:
2^6,3|a^2=e,b^2=ef,c^2=e,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 | #69248, #69309, C2×#4853, … |
| maximal normal subgroup of | #69248, #69309, C2×#4853, … |
| maximal quotient of | #68517, #68522, #68524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 412);