Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#2196 (order 1536) · Inn(G) = #937 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#124 ⋊ C2 |
| Sylow | C3 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #937 |
| commutator, cosocle | C2×C3×C4 | C23 |
| Frattini | C2×C4 | C22×D3 |
| Fitting | C3×#124 | C2 |
| socle | C2×C3 | #120 |
| hypercenter | #124 | D3 |
| hypocenter | C3 | #454 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #124 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,10)(7,16)(9,17)(11,12)(13,15)(18,19);(4,6,8,14)(5,9,12,19)(7,15,16,13)(10,18,11,17);(2,3)(4,5)(6,9)(7,10)(8,12)(11,16)(13,17)(14,19)(15,18)Polycyclic presentation:
2^6,3|b^2=f,d^2=f,c^a=cd,d^a=df,e^a=ef,g^a=g^2,c^b=ce,d^b=df,d^c=df30 conjugacy classes — view the full character table.
| maximal subgroup of | #67460, #67542, #69160, … |
| maximal normal subgroup of | #67460, #67542, #69160, … |
| maximal quotient of | #69543, #69562, #69563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 621);