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C₂²×C₃⋊#170 #91994

order	480
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×C₃⋊#170

Splittings

derived, Hall	C₃×C₅ ⋊ C₂²×C₈
Sylow	C₅ ⋊ C₂²×C₃⋊C₈
Sylow	C₃ ⋊ C₂²×#170

Subgroups

	subgroup	G/H
center, hypercenter	C₂³	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₂²×C₈
Frattini	C₂	C₂²×ΓL(1,𝔽₁₆)
Fitting, socle, prime radical	C₂³×C₃×C₅	C₄
cosocle	C₃×Dic₅	C₂³
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂³

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:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11746570, #11746789, C₂×Dic₃×#170, …
maximal normal subgroup of	#11746570, #11746789, C₂×Dic₃×#170, …
maximal quotient of	C₂²×C₃⋊#674, C₂×#91227, C₂×C₄×C₃⋊#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1070);