Dic₁₀₀ #87710

order	400
isoclinism	isoclinic class of Dic₁₀₀
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₀

Splittings

Sylow

C₂₅ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₁₀₀
commutator, cosocle	C₄×C₂₅	C₂²
Frattini	C₄×C₅	C₂×D₅
Fitting	C₈×C₂₅	C₂
socle, prime radical	C₂×C₅	D₂₀
hypercenter	C₈	D₂₅
hypocenter	C₂₅	Q₁₆
LCS	G ▷ C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

27 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 2 generators on 41 points:

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

Polycyclic presentation:

2^4,5^2|a^2=d,b^2=cd,c^2=d,e^5=f^4,b^a=bc,c^a=cd,e^a=e^4f,f^a=f^4

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11716348, Dic₂₀₀, #11716375, …
maximal normal subgroup of	#11716348, Dic₂₀₀, #11716375, …
maximal quotient of	#11716365, #11716367, C₂×Dic₁₀₀, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 4);

Dic100 #87710