Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×S42 (order 1152) · Inn(G) = C22×A4 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #585 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C23×C3 |
| Frattini, socle | C2 | C23×A4 |
| Fitting | #585 | C3 |
| hypercenter | #41 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C4 ◁ #41 | |
195 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8)(4,5,14)(7,18)(9,22,23,21,10,24)(11,12,25)(15,17,16)(19,30,31,29,20,32)(26,28,27);(1,25)(2,31)(3,16)(4,28)(5,11)(6,27)(7,24)(8,32)(9,30)(10,19)(12,15)(13,14)(17,26)(18,23)(20,21)(22,29);(1,2)(3,7)(4,9)(5,10)(6,8)(11,19)(12,20)(13,18)(14,23)(15,21)(16,24)(17,22)(25,31)(26,29)(27,32)(28,30)Polycyclic presentation:
2^3,3,2^3|b^2=g,e^2=g,f^2=g,c^a=cg,e^d=f,f^d=ef,f^e=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #73157, #2901⋊C3, #85541, … |
| maximal normal subgroup of | #73157, #2901⋊C3, #85541, … |
| maximal quotient of | #3428⋊C3, C4×#994, C4×2−1+4⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1504);