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) = C23 |
No aliases for this group.
| Fitting | C3×#102 ⋊ 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)(4,5,6,7)(9,14)(10,17)(12,20)(13,21)(16,23)(18,19);(8,10)(9,13)(11,18)(12,17)(14,16)(15,23)(19,20)(21,22);(2,3)(5,7)(8,9)(10,16)(11,14)(12,15)(13,18)(17,21)(19,23)(20,22)Polycyclic presentation:
2^6,3|c^2=df,e^2=f,b^a=bf,c^a=cd,g^a=g^2,c^b=ce,e^b=ef,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4783, #78747, #78759, … |
| maximal normal subgroup of | C2×#4783, #78747, #78759, … |
| maximal quotient of | #68202, #68203, #68204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 342);