C₂²⋊#261 #6123

order	208
isoclinism	isoclinic class of #6118
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₃) (order 1248) · Inn(G) = C₂×#261 · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product	C₂²⋊#261
unambiguous semidirect product	(C₂×#261)⋊C₂

Fitting	C₂²×C₁₃ ⋊ C₄
Sylow	C₁₃ ⋊ C₂²⋊C₄

34 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:

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

Polycyclic presentation:

2^4,13|a^2=c,b^a=bd,e^a=e^8,e^c=e^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(208, 34);