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.
| Sylow | C3 ⋊ #497 |
124 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 124 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)(9,14)(10,16)(11,19)(13,20)(17,18)(21,23);(5,7)(8,10,12,18)(9,13,14,21)(11,16,19,17)(15,23,22,20);(2,3)(4,5,6,7)(8,9)(10,13)(11,15)(12,14)(16,23)(17,20)(18,21)(19,22)Polycyclic presentation:
2^6,3|a^2=e,b^2=f,d^2=f,b^a=be,c^a=cef,g^a=g^2,c^b=cd,d^b=df,d^c=df30 conjugacy classes — view the full character table.
| maximal subgroup of | #79640, #79642, #79665, … |
| maximal normal subgroup of | #79640, #79642, #79665, … |
| maximal quotient of | #70159, #70183, #70192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 726);