Dic₂₆ #1058

order	104
isoclinism	isoclinic class of Dic₂₆
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₃) (order 1248) · Inn(G) = C₂×D₁₃ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₆
unambiguous semidirect product	C₁₃⋊Q₈

C₁₃ ⋊ Q₈

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₃
commutator, socle, prime radical, cosocle	C₂×C₁₃	C₂²
Fitting	C₄×C₁₃	C₂
hypercenter	C₄	D₁₃
hypocenter	C₁₃	Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(104, 4);