Dic₁₄ #284

order	56
isoclinism	isoclinic class of Dic₁₄
automorphism	Aut(G) = D₄×AGL(1,𝔽₇) (order 336) · Inn(G) = C₂×D₇ · Out(G) = C₂²×C₃
holomorph	G ⋊ Aut(G) = #72215081
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₁₄
unambiguous semidirect product	C₇⋊Q₈

C₇ ⋊ Q₈

	subgroup	G/H
center, Frattini	C₂	C₂×D₇
commutator, socle, prime radical, cosocle	C₂×C₇	C₂²
Fitting	C₄×C₇	C₂
hypercenter	C₄	D₇
hypocenter	C₇	Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

2^3,7|a^2=c,b^2=c,b^a=bc,d^a=d^6

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(56, 3);