Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#612424 (order 3072) · Inn(G) = C2×#67 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #480 |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(13,16)(14,19)(17,18);(5,8)(9,11);(2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,16)(15,18)(17,19)Polycyclic presentation:
2^6,3|c^2=e,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #69708, #69709, #69711, … |
| maximal normal subgroup of | #69708, #69709, #69711, … |
| maximal quotient of | #69458, #69462, #69518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 526);