Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#7630939 (order 3072) · Inn(G) = D3×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×C3×SD16 ⋊ C2 |
| Sylow | C3 ⋊ #495 |
130 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(5,7)(8,10,12,18)(9,14,13,16)(11,20,19,17)(15,22,21,23);(5,7)(9,13)(10,17)(11,19)(14,22)(16,23)(18,20);(2,3)(4,5,6,7)(8,9)(10,16)(11,15)(12,13)(14,18)(17,22)(19,21)(20,23)Polycyclic presentation:
2^6,3|a^2=e,c^2=f,d^2=f,b^a=bef,c^a=cef,g^a=g^2,c^b=cd,d^b=df,d^c=df36 conjugacy classes — view the full character table.
| maximal subgroup of | #79638, #79639, #79663, … |
| maximal normal subgroup of | #79638, #79639, #79663, … |
| maximal quotient of | #70158, #70182, #70190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 727);