Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C23×D3×#120 (order 1536) · Inn(G) = C22×D3 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ C8×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D3 |
| commutator | C2×C3 | C22×C8 |
| Fitting | C3×C4⋊C8 | C2 |
| socle, prime radical | C22×C3 | C22×C4 |
| cosocle | C2×C3×C4 | C23 |
| hypercenter | C4⋊C8 | D3 |
| hypocenter | C3 | C8×Q8 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C4⋊C8 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,7,11,8,5,9,10)(14,18)(16,19);(12,14,15,18)(13,16,17,19);(2,3)(4,5,7,10,8,6,9,11)(12,13,15,17)(14,16,18,19)Polycyclic presentation:
2^6,3|a^2=e,b^2=d,c^2=de,e^2=f,g^a=g^2,c^b=cd60 conjugacy classes — view the full character table.
| maximal subgroup of | #68149, #68150, #68155, … |
| maximal normal subgroup of | #68149, #68150, #68155, … |
| maximal quotient of | #68609, #68619, #68651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 389);