Dic₃₄ #3625

order	136
isoclinism	isoclinic class of Dic₃₄
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₇) (order 2176) · Inn(G) = C₂×D₁₇ · Out(G) = C₂×C₁₆
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₄
unambiguous semidirect product	C₁₇⋊Q₈

C₁₇ ⋊ Q₈

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₇
commutator, socle, prime radical, cosocle	C₂×C₁₇	C₂²
Fitting	C₄×C₁₇	C₂
hypercenter	C₄	D₁₇
hypocenter	C₁₇	Q₈
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 136

A minimal faithful permutation representation — 2 generators on 25 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25);(1,5,3,7)(2,6,4,8)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(136, 4);