Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4625 |
| automorphism | Aut(G) = #4625⋊C2 (order 384) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C22⋊A4 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C22⋊A4 | C4 |
| Frattini | C23 | S4 |
| Fitting | #120 | D3 |
| socle | C22 | SL(2,ℤ/4) |
| prime radical, cosocle | #120⋊C3 | C2 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,6,16,7,5,9,8,3)(2,15,11,13,10,12,14,4);(2,7,16)(3,12,9)(4,5,13)(6,14,8)(10,11,15)Polycyclic presentation:
2^2,3,2^4|a^2=b,c^a=c^2,d^a=e,e^a=dfg,f^a=fg,d^b=dfg,e^b=ef,d^c=eg,e^c=def,f^c=g,g^c=fg| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 3 | 4 | 12 | 12 | 24 | 24 | 24 | 24 | 32 | 32 |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 3 | 6 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 |
| χ3 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | ||||
| χ4 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | ||||
| χ5 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 |
| χ6 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 |
| χ7 | 3 | 3 | 3 | -1 | -1 | 1 | 1 | -1 | -1 | 0 | 0 |
| χ8 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 |
| χ9 | 3 | 3 | -3 | -1 | 1 | 0 | 0 | ||||
| χ10 | 3 | 3 | -3 | -1 | 1 | 0 | 0 | ||||
| χ11 | 12 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | C2×#4625, #4625⋊C2, C3×#4625, … |
| maximal normal subgroup of | C2×#4625, #4625⋊C2, C3×#4625, … |
| maximal quotient of | #67998, #68001, #68002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 184);