Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×#120 (order 3072) · Inn(G) = C2×D12 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C4⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #460 |
128 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)(5,7)(8,10,11,9,12,13,15,14);(4,5,6,7);(2,3)(8,9,12,14)(10,15,13,11)Polycyclic presentation:
2^6,3|a^2=f,b^2=e,c^2=d,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce39 conjugacy classes — view the full character table.
| maximal subgroup of | #68186, #68187, #68198, … |
| maximal normal subgroup of | #68186, #68187, #68198, … |
| maximal quotient of | #68546, #68550, #68551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 407);