Dic₁₀₈ #88357

order	432
isoclinism	isoclinic class of Dic₁₀₈
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₈

Splittings

Sylow

C₂₇ ⋊ Q₁₆

Subgroups

	subgroup	G/H
center	C₂	D₁₀₈
commutator, cosocle	C₄×C₂₇	C₂²
Frattini	C₄×C₉	C₂×D₃
Fitting	C₈×C₂₇	C₂
socle, prime radical	C₂×C₃	D₃₆
hypercenter	C₈	D₂₇
hypocenter	C₂₇	Q₁₆
LCS	G ▷ C₄×C₂₇ ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 2 generators on 43 points:

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

Polycyclic presentation:

2^4,3^3|a^2=d,b^2=cd,c^2=d,e^3=f^2g,f^3=g^2,b^a=bc,c^a=cd,e^a=e^2f,f^a=f^2g,g^a=g^2

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₂₁₆, #11720041, #11720066, …
maximal normal subgroup of	Dic₂₁₆, #11720041, #11720066, …
maximal quotient of	#11720057, #11720059, C₂×Dic₁₀₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 4);

Dic108 #88357