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C₂²×C₃×#170 #91981

order	480
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₈ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×C₃×#170

Splittings

derived, Sylow	C₅ ⋊ C₂²×C₃×C₈
Hall	C₃×C₅ ⋊ C₂²×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₃	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂²×C₃×C₈
Frattini	C₂	C₂²×C₃×AGL(1,𝔽₅)
Fitting, socle, prime radical	C₂³×C₃×C₅	C₄
cosocle	Dic₅	C₂³×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂³×C₃

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11746570, C₃×#64918, C₂×Dic₃×#170, …
maximal normal subgroup of	#11746570, C₃×#64918, C₂×Dic₃×#170, …
maximal quotient of	C₂²×C₃×#674, C₂×C₃×#3998, C₂×C₃×C₄×#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1057);