C₄⋊#595 #63245

order	272
isoclinism	isoclinic class of #63243
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₇) (order 2176) · Inn(G) = C₂×#595 · Out(G) = C₂×C₈
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

Fitting	C₄×C₁₇ ⋊ C₄
Sylow	C₁₇ ⋊ C₄⋊C₄

26 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 68

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,5,10,6,11,15,17,13,8,14,16,12,3,7,9,4)(18,19,20,21);(2,3,4,6)(5,8,9,13)(7,12,10,11)(14,16,17,15)(19,21)

Polycyclic presentation:

2^4,17|a^2=c,b^2=d,b^a=bd,e^a=e^13,e^c=e^16

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(272, 32);