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 | C3×#103 ⋊ C2 |
| Sylow | C3 ⋊ #467 |
114 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 114 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)(4,5)(6,7)(8,11,13,10)(9,14,16,18)(12,17,19,20)(15,22,23,21);(5,7)(8,10,12,17,13,11,19,20)(9,14,15,21,16,18,23,22);(2,3)(5,7)(8,9)(10,14)(11,18)(12,15)(13,16)(17,21)(19,23)(20,22)Polycyclic presentation:
2^6,3|b^2=d,c^2=f,d^2=f,e^2=f,c^a=cde,g^a=g^2,c^b=cef,d^c=df,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4813, #78800, #78806, … |
| maximal normal subgroup of | C2×#4813, #78800, #78806, … |
| maximal quotient of | #68441, #68447, #68465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 372);