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) = C4 |
No aliases for this group.
| Fitting | C3×C4⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #489 |
100 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)(4,6,7,11)(5,9,8,10)(13,18)(14,15)(17,19);(4,5,7,8)(6,10,11,9)(12,14,16,15)(13,17,19,18);(2,3)(5,8)(9,10)(12,13)(14,17)(15,18)(16,19)Polycyclic presentation:
2^6,3|b^2=ef,c^2=e,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,c^b=cef,d^c=df30 conjugacy classes — view the full character table.
| maximal subgroup of | #71244, #71246, #71251, … |
| maximal normal subgroup of | #71244, #71246, #71251, … |
| maximal quotient of | #69575, #69581, #69585, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 648);