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×D16 ⋊ C2 |
| Sylow | C3 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D8 |
| commutator, cosocle | C3×C8 | C23 |
| Frattini | C8 | C22×D3 |
| Fitting | C3×D16 | C2 |
| socle | C2×C3 | C2×D8 |
| hypercenter | D16 | D3 |
| hypocenter | C3 | #508 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D16 | |
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)(5,11)(6,15)(7,19)(8,20)(10,23)(12,29)(14,28)(16,17)(18,34)(21,35)(22,33)(24,31)(25,32)(26,30);(4,6)(5,10)(7,15)(8,22)(9,18)(11,14)(12,28)(13,31)(16,19)(17,20)(21,34)(23,26)(24,30)(25,27)(29,32)(33,35);(2,3)(4,5)(6,14)(7,12)(8,13)(9,11)(10,18)(15,23)(16,32)(17,31)(19,30)(20,27)(21,26)(22,25)(24,33)(28,34)(29,35)Polycyclic presentation:
2^6,3|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 | #69376, #69377, #69379, … |
| maximal normal subgroup of | #69376, #69377, #69379, … |
| maximal quotient of | #69175, #69180, #69184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 471);