Finite Group DatabasebetaFinite Group Databasebeta| order | 48 |
| automorphism | Aut(G) = C2×D3×S4 (order 288) · Inn(G) = C22×D3 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×Q8 ⋊ C2 |
| Sylow | C3 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3 |
| commutator, socle, cosocle | C2×C3 | C23 |
| Fitting | C3×Q8 | C2 |
| hypercenter | Q8 | D3 |
| hypocenter | C3 | #41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 11 points:
(1,2,3)(4,6,7,10)(5,8,9,11);(4,5,7,9)(6,11,10,8);(2,3)(6,10)(8,11)Polycyclic presentation:
2^4,3|b^2=d,c^2=d,c^a=cd,e^a=e^2,c^b=cd15 conjugacy classes — view the full character table.
| maximal subgroup of | #914, #916, #918, … |
| maximal normal subgroup of | #914, #916, #918, … |
| maximal quotient of | #890, #892, #893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(48, 41);