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Dic₉₀ #67095

order	360
isoclinism	isoclinic class of Dic₉₀
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

Dic₉₀

Splittings

Hall	C₅×C₉ ⋊ Q₈
Sylow	C₉ ⋊ Dic₁₀
Sylow	C₅ ⋊ Dic₁₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄₅
commutator, cosocle	C₂×C₅×C₉	C₂²
Frattini	C₂×C₃	C₂×D₁₅
Fitting	C₄×C₅×C₉	C₂
socle, prime radical	C₂×C₃×C₅	C₂×D₃
hypercenter	C₄	D₄₅
hypocenter	C₅×C₉	Q₈
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17)(18,19,20,21,22);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)(19,22)(20,21)

Polycyclic presentation:

2^3,3,5,3|a^2=c,b^2=c,d^3=f^2,b^a=bc,d^a=d^2f,e^a=e^4,f^a=f^2

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10623542, #10623545, #10623551, …
maximal normal subgroup of	#10623542, #10623545, #10623551, …
maximal quotient of	#10623604, C₄⋊Dic₄₅, C₂×Dic₉₀, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 25);