Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×#120 (order 3072) · Inn(G) = C2×#67 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #428 |
118 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)(8,9)(10,15)(11,13)(12,14);(9,13)(12,15);(2,3)(4,5,7,6)(9,12)(10,14)(13,15)Polycyclic presentation:
2^6,3|a^2=e,c^2=e,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=cf,d^c=df42 conjugacy classes — view the full character table.
| maximal subgroup of | #68733, #68734, #68735, … |
| maximal normal subgroup of | #68733, #68734, #68735, … |
| maximal quotient of | #69466, #69467, #69481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 554);