Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = D3×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | C3×#102 ⋊ C2 |
| Sylow | C3 ⋊ #482 |
106 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,9)(8,11)(13,17)(15,19)(16,18);(4,6,7,10)(5,8,9,11)(12,13,14,16)(15,17,19,18);(2,3)(4,5)(6,8)(7,9)(10,11)(13,16)(17,18)Polycyclic presentation:
2^6,3|b^2=d,e^2=f,b^a=bf,c^a=cdf,g^a=g^2,c^b=ce,e^b=ef,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4784, #78756, #78759, … |
| maximal normal subgroup of | C2×#4784, #78756, #78759, … |
| maximal quotient of | #68789, #68812, #68820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 343);