Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#23900 (order 3072) · Inn(G) = C22×D3 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #424 |
114 conjugacy classes of subgroups.
| 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)(8,10,11,17,12,16,18,22)(9,14,15,21,13,19,20,23);(9,13)(10,16)(15,20)(17,22);(2,3)(4,5,6,7)(8,9)(10,14)(11,15)(12,13)(16,19)(17,21)(18,20)(22,23)Polycyclic presentation:
2^6,3|a^2=e,c^2=d,d^2=f,b^a=bf,c^a=ce,g^a=g^2,c^b=cf48 conjugacy classes — view the full character table.
| maximal subgroup of | #67533, #68698, #68718, … |
| maximal normal subgroup of | #67533, #68698, #68718, … |
| maximal quotient of | #69832, #69836, #69909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 678);