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(C₂²⋊C₉)⋊AGL(1,𝔽₅) #10623923

order	720
isoclinism	isoclinic class of (C₂²⋊C₉)⋊AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

(C₂²⋊C₉)⋊AGL(1,𝔽₅)

Splittings

derived	C₅×C₂²⋊C₉ ⋊ C₄
Sylow	C₅ ⋊ (C₂²⋊C₉)⋊C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₅×C₂²⋊C₉	C₄
Frattini	C₃	A₄⋊AGL(1,𝔽₅)
Fitting, socle	C₂²×C₃×C₅	Dic₃
prime radical	C₂²×C₃×D₅	D₃
cosocle	D₅×C₂²⋊C₉	C₂
LCS	G ▷ C₅×C₂²⋊C₉
UCS	C₁

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 90

Polycyclic presentation:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×(C₂²⋊C₉)⋊C₂, C₂×(C₂²⋊C₉)⋊AGL(1,𝔽₅), #72247667, …
maximal normal subgroup of	AGL(1,𝔽₅)×(C₂²⋊C₉)⋊C₂, C₂×(C₂²⋊C₉)⋊AGL(1,𝔽₅), #72247678
maximal quotient of	#13085140, #13085141, (C₂²⋊C₉)⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 392);