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C₃⋊(C₉⋊#170) #11761725

order	1080
isoclinism	isoclinic class of C₃⋊(C₉⋊AGL(1,𝔽₅))
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product	C₃⋊(C₉⋊#170)
unambiguous semidirect product	C₉⋊(C₃⋊#170)

Splittings

derived, Hall	C₃×C₅×C₉ ⋊ C₈
Sylow	C₃×C₉ ⋊ #170
Sylow	C₅ ⋊ C₃⋊(C₉⋊C₈)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊(C₉⋊AGL(1,𝔽₅))
commutator, hypocenter	C₃×C₅×C₉	C₈
Frattini	C₂×C₃	C₃⋊ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₅×C₉	C₄
socle, prime radical	C₂×C₃²×C₅	Dic₃
cosocle	C₃×C₉×Dic₅	C₂
LCS	G ▷ C₃×C₅×C₉
UCS	C₁ ◁ C₂

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 1080

Polycyclic presentation:

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

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1080, 71);