C₄⋊(D₁₃⋊C₃) #64594

order	312
isoclinism	isoclinic class of (C₁₃⋊C₃)⋊Q₈
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₃) (order 1248) · Inn(G) = C₂×D₁₃⋊C₃ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₄⋊(D₁₃⋊C₃)
unambiguous semidirect product	D₅₂⋊C₃

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 52

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,8)(3,9,13)(4,6,11)(5,12,7)(14,15,16,17);(2,3)(4,6)(5,9)(7,8)(10,12)(11,13)(15,17)

Polycyclic presentation:

2^2,3,2,13|b^2=d,b^a=bd,e^a=e^12,e^c=e^3

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 10);