Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #826 |
| automorphism | Aut(G) = C2×D3×#442 (order 768) · Inn(G) = #213 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#135 ⋊ C2 |
| Sylow | C3 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #213 |
| commutator, cosocle | C3×C8 | C23 |
| Frattini | C8 | C22×D3 |
| Fitting | C3×#135 | C2 |
| socle | C2×C3 | C2×D8 |
| hypercenter | #135 | D3 |
| hypocenter | C3 | #508 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #135 | |
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)(7,18)(8,19)(10,22)(12,13)(14,27)(15,28)(16,29)(20,32)(21,26)(23,24)(25,34)(30,35)(31,33);(4,6,9,17)(5,10,14,25)(7,15,20,30)(8,16,19,29)(11,22,27,34)(12,23,26,33)(13,24,21,31)(18,28,32,35);(2,3)(4,5)(6,10)(7,11)(8,21)(9,14)(12,18)(13,19)(15,22)(16,31)(17,25)(20,27)(23,28)(24,29)(26,32)(30,34)(33,35)Polycyclic presentation:
2^6,3|b^2=f,d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^2,d^c=de,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69238, #69239, #70921, … |
| maximal normal subgroup of | #69238, #69239, #70921, … |
| maximal quotient of | #70648, #70651, #70653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 753);