Dic₅₂ #6097

order	208
isoclinism	isoclinic class of #6095
automorphism	Aut(G) = AGL(1,ℤ/8)×AGL(1,𝔽₁₃) (order 4992) · Inn(G) = D₅₂ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₂

Splittings

Sylow

C₁₃ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₅₂
commutator, cosocle	C₄×C₁₃	C₂²
Frattini	C₄	C₂×D₁₃
Fitting	C₈×C₁₃	C₂
socle, prime radical	C₂×C₁₃	D₄
hypercenter	C₈	D₁₃
hypocenter	C₁₃	Q₁₆
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 208

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23,24,25,26,27,28,29);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Polycyclic presentation:

2^4,13|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#88040, Dic₁₀₄, #88067, …
maximal normal subgroup of	#88040, Dic₁₀₄, #88067, …
maximal quotient of	#88056, #88058, C₂×Dic₅₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(208, 8);

Dic52 #6097