Dic₄₄ #4316

order	176
isoclinism	isoclinic class of #4314
automorphism	Aut(G) = AGL(1,ℤ/8)×AGL(1,𝔽₁₁) (order 3520) · Inn(G) = D₄₄ · Out(G) = C₂³×C₅
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₄

Splittings

Sylow

C₁₁ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₄₄
commutator, cosocle	C₄×C₁₁	C₂²
Frattini	C₄	C₂×D₁₁
Fitting	C₈×C₁₁	C₂
socle, prime radical	C₂×C₁₁	D₄
hypercenter	C₈	D₁₁
hypocenter	C₁₁	Q₁₆
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 176

A minimal faithful permutation representation — 2 generators on 27 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,27)(19,26)(20,25)(21,24)(22,23)

Polycyclic presentation:

2^4,11|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^10

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#66864, Dic₈₈, #66891, …
maximal normal subgroup of	#66864, Dic₈₈, #66891, …
maximal quotient of	#66880, #66882, C₂×Dic₄₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(176, 7);

Dic44 #4316