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) = C22 |
No aliases for this group.
| Fitting | C2×C3×Q16 ⋊ C2 |
| Sylow | C3 ⋊ #464 |
112 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 41 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)(8,10,12,16)(9,13,15,22)(11,20,21,18)(14,23,19,17);(8,9,12,15)(10,17,16,23)(11,19,21,14)(13,20,22,18);(2,3)(4,5,6,7)(10,16)(13,22)(17,23)(18,20)Polycyclic presentation:
2^6,3|a^2=e,b^2=f,c^2=f,d^2=f,c^a=cef,g^a=g^2,c^b=cd,d^b=df,d^c=df33 conjugacy classes — view the full character table.
| maximal subgroup of | #79700, #79701, #79703, … |
| maximal normal subgroup of | #79700, #79701, #79703, … |
| maximal quotient of | #70222, #70230, #70234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 744);