Dic₁₀₄ #88041

order	416
isoclinism	isoclinic class of D₂₀₈
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₄

Splittings

Sylow

C₁₃ ⋊ Q₃₂

Subgroups

	subgroup	G/H
center	C₂	D₁₀₄
commutator, cosocle	C₈×C₁₃	C₂²
Frattini	C₈	C₂×D₁₃
Fitting	C₁₃×C₁₆	C₂
socle, prime radical	C₂×C₁₃	D₈
hypercenter	C₁₆	D₁₃
hypocenter	C₁₃	Q₃₂
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(33,34,35,36,37,38,39,40,41,42,43,44,45);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Polycyclic presentation:

2^5,13|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^12

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11718041, Dic₂₀₈, #11718112, …
maximal normal subgroup of	#11718041, Dic₂₀₈, #11718112, …
maximal quotient of	#11718095, #11718096, C₂×Dic₁₀₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 8);

Dic104 #88041