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C₉⋊#170 #67076

order	360
isoclinism	isoclinic class of C₉⋊AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

Splittings

derived, Hall	C₅×C₉ ⋊ C₈
Sylow	C₉ ⋊ #170
Sylow	C₅ ⋊ C₉⋊C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₉⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₉	C₈
Frattini	C₂×C₃	ΓL(1,𝔽₁₆)
Fitting	C₂×C₅×C₉	C₄
socle, prime radical	C₂×C₃×C₅	Dic₃
cosocle	C₉×Dic₅	C₂
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₂

24 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×#170, #10623627, #10623628, …
maximal normal subgroup of	D₉×#170, #10623627, #10623628, …
maximal quotient of	C₉⋊#674, C₂×C₉⋊#170, C₂₇⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 6);