D₁₅⋊Q₈ #6770

order	240
automorphism	Aut(G) = C₂×D₃×D₄×AGL(1,𝔽₅) (order 1920) · Inn(G) = C₂×D₃×D₅ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

D₁₅⋊Q₈

76 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2,4,5,3)(6,7,8)(9,12,13,16)(10,14,11,15);(7,8)(9,11,13,10)(12,14,16,15);(2,3)(4,5)(9,10,13,11)(12,15,16,14)

Polycyclic presentation:

2^4,3,5|a^2=d,b^2=d,c^2=d,c^a=cd,f^a=f^4,c^b=cd,e^b=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 131);