Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| isoclinism | isoclinic class of #857 |
| automorphism | Aut(G) = #857⋊C2 (order 192) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C42 ⋊ D3 |
| derived | C42⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C42⋊C3 | C2 |
| Frattini, socle | C22 | S4 |
| Fitting | C42 | D3 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,3,4,8)(2,12,11,10,6,9,7,5);(1,2,5)(3,7,9)(4,6,10)(8,11,12)Polycyclic presentation:
2,3,2^4|c^2=ef,d^2=e,b^a=b^2,c^a=d,d^a=c,e^a=ef,c^b=d,d^b=cdf,e^b=f,f^b=ef| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 3 | 3 | 3 | 6 | 12 | 12 | 12 | 12 | 32 |
| order | 1 | 2 | 4 | 4 | 4 | 2 | 4 | 8 | 8 | 3 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 |
| χ3 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 |
| χ4 | 3 | 3 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 0 |
| χ5 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 |
| χ6 | 3 | -1 | 1 | 1 | -1 | 0 | ||||
| χ7 | 3 | -1 | 1 | -1 | 1 | 0 | ||||
| χ8 | 3 | -1 | 1 | 1 | -1 | 0 | ||||
| χ9 | 3 | -1 | 1 | -1 | 1 | 0 | ||||
| χ10 | 6 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | C2×#857, #857⋊C2, C3×#857, … |
| maximal normal subgroup of | C2×#857, #857⋊C2, C3×#857, … |
| maximal quotient of | #4621, #4622, #4623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 64);