Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×AGL(1,ℤ/16) (order 1536) · Inn(G) = D3×D8 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×Q32 ⋊ C2 |
| Sylow | C3 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D8 |
| commutator, cosocle | C3×C8 | C23 |
| Frattini | C8 | C22×D3 |
| Fitting | C3×Q32 | C2 |
| socle | C2×C3 | C2×D8 |
| hypercenter | Q32 | D3 |
| hypocenter | C3 | #508 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q32 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,6,9,14)(5,11,10,25)(7,19,22,35)(8,24,20,17)(12,30,26,34)(13,31,23,28)(15,33,32,18)(16,29,27,21);(4,5,9,10)(6,15,14,32)(7,18,22,33)(8,23,20,13)(11,27,25,16)(12,29,26,21)(17,34,24,30)(19,31,35,28);(2,3)(5,10)(6,14)(12,26)(13,23)(16,27)(17,24)(18,33)(19,35)Polycyclic presentation:
2^6,3|b^2=f,c^2=f,d^2=ef,e^2=f,b^a=bf,c^a=cf,g^a=g^2,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | #69379, #69380, #69383, … |
| maximal normal subgroup of | #69379, #69380, #69383, … |
| maximal quotient of | #69206, #69213, #69214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 478);