Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C22×D3×#3003 (order 3072) · Inn(G) = D3×D4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#107 ⋊ C2 |
| Sylow | C3 ⋊ C4×D8 |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(4,6,7,5)(9,12)(10,15)(13,14);(4,5,7,6)(8,9)(10,12)(11,14)(13,15);(2,3)(4,5,7,6)Polycyclic presentation:
2^6,3|a^2=e,b^2=e,c^2=e,d^2=f,g^a=g^2,c^b=cd,d^b=df,d^c=df42 conjugacy classes — view the full character table.
| maximal subgroup of | #69174, #69177, #69202, … |
| maximal normal subgroup of | #69174, #69177, #69202, … |
| maximal quotient of | #68556, #68559, #68563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 431);