Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#442 (order 768) · Inn(G) = D3×D8 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×SD32 ⋊ C2 |
| Sylow | C3 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D8 |
| commutator, cosocle | C3×C8 | C23 |
| Frattini | C8 | C22×D3 |
| Fitting | C3×SD32 | C2 |
| socle | C2×C3 | C2×D8 |
| hypercenter | SD32 | D3 |
| hypocenter | C3 | #508 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ SD32 | |
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)(4,6,9,17)(5,11,10,14)(7,19,21,34)(8,22,20,16)(12,29,23,35)(13,31,24,27)(15,32,33,18)(25,30,26,28);(5,10)(6,15)(7,18)(8,20)(11,25)(12,28)(14,26)(16,34)(17,33)(19,22)(21,32)(23,30)(27,29)(31,35);(2,3)(4,5)(6,14)(7,12)(8,13)(9,10)(11,17)(15,25)(16,31)(18,30)(19,35)(20,24)(21,23)(22,27)(26,33)(28,32)(29,34)Polycyclic presentation:
2^6,3|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 | C2×#4916, #82024, D3×#508, … |
| maximal normal subgroup of | C2×#4916, #82024, D3×#508, … |
| maximal quotient of | #69174, #69181, #69182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 475);