Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×S4×AGL(1,ℤ/8) (order 4608) · Inn(G) = C22×D12 · Out(G) = C23×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#131 ⋊ C2 |
| Sylow | C3 ⋊ #578 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D12 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C3×#131 | C2 |
| socle | C2×C3 | C22×D4 |
| hypercenter | #131 | D3 |
| hypocenter | C3 | #578 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 | |
248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,8,7,19,10,22,9,21)(5,13,23,33,14,28,12,20)(6,16,18,32,17,31,15,29)(11,25,24,34,27,35,26,30);(4,7,10,9)(5,12,14,23)(6,15,17,18)(8,19,22,21)(11,24,27,26)(13,20,28,33)(16,29,31,32)(25,34,35,30);(4,6,10,17)(5,11,14,27)(7,18,9,15)(8,16,22,31)(12,26,23,24)(13,25,28,35)(19,32,21,29)(20,30,33,34);(2,3)(4,5)(6,11)(7,12)(8,20)(9,23)(10,14)(13,21)(15,24)(16,30)(17,27)(18,26)(19,28)(22,33)(25,29)(31,34)(32,35)Polycyclic presentation:
2^6,3|b^2=f,c^2=f,d^2=ef,e^2=f,d^a=de,e^a=ef,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #69127, #69129, #71095, … |
| maximal normal subgroup of | #69127, #69129, #71095, … |
| maximal quotient of | #78546, #78562, #78568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1312);