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) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C4⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #476 |
108 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)(5,8,9,10)(13,18)(14,15)(17,19);(4,5,7,9)(6,10,11,8)(12,14,16,15)(13,17,19,18);(2,3)(12,13)(14,17)(15,18)(16,19)Polycyclic presentation:
2^6,3|b^2=e,c^2=ef,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce,d^c=df33 conjugacy classes — view the full character table.
| maximal subgroup of | #68496, #68497, #68498, … |
| maximal normal subgroup of | #68496, #68497, #68498, … |
| maximal quotient of | #69568, #69572, #69584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 643);