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×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #446 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D3 |
| commutator, socle, prime radical | C22×C3 | C22×C4 |
| Fitting | C3×C4⋊C8 | C2 |
| cosocle | C2×C3×C4 | C23 |
| hypercenter | C4⋊C8 | D3 |
| hypocenter | C3 | #446 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C4⋊C8 | |
94 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)(6,10)(8,11)(12,14,15,17,16,18,19,13);(4,6,7,10)(5,8,9,11);(2,3)(4,5,7,9)(6,8,10,11)(12,13,15,14,16,17,19,18)Polycyclic presentation:
2^6,3|a^2=e,b^2=d,c^2=de,e^2=f,c^a=cf,g^a=g^2,c^b=cd48 conjugacy classes — view the full character table.
| maximal subgroup of | #68154, #68156, #68161, … |
| maximal normal subgroup of | #68154, #68156, #68161, … |
| maximal quotient of | #68612, #68620, #68653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 390);