Finite Group DatabasebetaFinite Group Databasebeta| order | 48 |
| isoclinism | isoclinic class of #226 |
| automorphism | Aut(G) = C2×S4 (order 48) · Inn(G) = S4 · Out(G) = C2 |
| holomorph | G ⋊ Aut(G) = #72215280 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | S4 |
| commutator, prime radical, cosocle, hypocenter | SL(2,𝔽3) | C2 |
| Fitting | Q8 | D3 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
13 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,6,10,13,5,15,12,7)(2,11,14,4,8,3,16,9);(3,9,12)(4,10,11)(6,13,16)(7,14,15)Polycyclic presentation:
2,3,2^3|a^2=e,c^2=e,d^2=e,b^a=b^2,c^a=d,d^a=c,c^b=de,d^b=cd,d^c=de| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 6 | 6 | 6 | 8 | 8 | 12 |
| order | 1 | 2 | 4 | 8 | 8 | 3 | 6 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 |
| χ3 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 0 |
| χ4 | 2 | -2 | 0 | -1 | 1 | 0 | ||
| χ5 | 2 | -2 | 0 | -1 | 1 | 0 | ||
| χ6 | 3 | 3 | -1 | 1 | 1 | 0 | 0 | -1 |
| χ7 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 1 |
| χ8 | 4 | -4 | 0 | 0 | 0 | 1 | -1 | 0 |
| maximal subgroup of | C2×#226, #983, #984, … |
| maximal normal subgroup of | C2×#226, #983, #984, … |
| maximal quotient of | #859, C2×#226, #3688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 28);