Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×D3×#2846 (order 1536) · Inn(G) = C2×D3 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×C4×C8 ⋊ C2 |
| Sylow | C3 ⋊ #405 |
130 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)(8,10,11,16,12,17,18,22)(9,14,15,21,13,19,20,23);(4,5,6,7)(9,13)(14,19)(15,20)(21,23);(2,3)(8,9)(10,14)(11,15)(12,13)(16,21)(17,19)(18,20)(22,23)Polycyclic presentation:
2^6,3|b^2=e,c^2=d,d^2=f,b^a=bf,g^a=g^272 conjugacy classes — view the full character table.
| maximal subgroup of | #69051, #69053, #69072, … |
| maximal normal subgroup of | #69051, #69053, #69072, … |
| maximal quotient of | C8×ΓL(1,𝔽25), #68047, #68048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 248);