C₃⋊#3633 #87997

order	408
isoclinism	isoclinic class of C₃⋊#3633
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₇) (order 1632) · Inn(G) = G · Out(G) = C₄
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₁₇	C₈
prime radical	C₃×D₁₇	C₄
cosocle	C₃×#595	C₂
LCS	G ▷ C₃×C₁₇
UCS	C₁

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 51

A minimal faithful permutation representation — 2 generators on 20 points:

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

Polycyclic presentation:

2^3,3,17|a^2=bc,b^2=c,d^a=d^2,e^a=e^2,e^b=e^13,e^c=e^16

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(408, 34);