Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4519 |
| automorphism | Aut(G) = D3×#12067 (order 1536) · Inn(G) = #826 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C3×Q32 ⋊ C2 |
| Sylow | C3 ⋊ SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #826 |
| commutator, cosocle | C3×C16 | C22 |
| Frattini | C16 | C2×D3 |
| Fitting | C3×Q32 | C2 |
| socle | C2×C3 | D16 |
| prime radical | D48 | C2 |
| hypercenter | Q32 | D3 |
| hypocenter | C3 | SD64 |
| LCS | G ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,5,9,13)(6,15,18,32)(7,19,21,35)(8,22,20,12)(10,23,25,14)(11,27,28,16)(17,34,30,26)(24,31,29,33);(2,3)(5,10)(6,14)(7,16)(8,20)(11,26)(12,29)(13,25)(15,19)(17,33)(18,23)(21,27)(22,24)(28,34)(30,31)(32,35)Polycyclic presentation:
2^6,3|b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,c^b=cd,d^b=de,e^b=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | D3×SD64, #69380, #69383, … |
| maximal normal subgroup of | D3×SD64, #69380, #69383, … |
| maximal quotient of | #67576, #67577, #67826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 80);