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) = C2 |
No aliases for this group.
| Fitting | C3×D16 ⋊ C2 |
| Sylow | C3 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #826 |
| commutator, cosocle | C3×C16 | C22 |
| Frattini | C16 | C2×D3 |
| Fitting | C3×D16 | C2 |
| socle | C2×C3 | D16 |
| hypercenter | D16 | D3 |
| hypocenter | C3 | D32 |
| LCS | G ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D16 | |
34 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)(5,10)(6,15)(7,16)(8,20)(11,12)(13,29)(14,25)(17,34)(18,32)(19,23)(21,33)(22,24)(26,35)(27,28)(30,31);(2,3)(4,5)(6,10)(7,19)(8,22)(9,14)(11,15)(12,16)(13,20)(17,29)(18,25)(21,35)(23,31)(24,30)(26,34)(27,32)(28,33)Polycyclic presentation:
2^6,3|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×D32, #69377, #69379, … |
| maximal normal subgroup of | D3×D32, #69377, #69379, … |
| maximal quotient of | #67575, #67577, #67607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 78);