Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = C2×#67 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C22⋊Q8 ⋊ C2 |
| Sylow | C3 ⋊ #452 |
152 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 152 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)(8,11,13,19)(9,15,16,23)(10,17,12,21)(14,22,20,18);(4,5)(6,7)(8,10,13,12)(9,14,16,20)(11,17,19,21)(15,22,23,18);(2,3)(8,9)(10,14)(11,18)(12,20)(13,16)(15,17)(19,22)(21,23)Polycyclic presentation:
2^6,3|b^2=f,c^2=f,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 | #80142, #80143, #80148, … |
| maximal normal subgroup of | #80142, #80143, #80148, … |
| maximal quotient of | #69506, #69522, #69523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 606);