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C₃×C₄⋊#170 #91205

order	480
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₃×C₄⋊#170
unambiguous semidirect product	C₄⋊(C₃×#170)

Splittings

Hall	C₃×C₅ ⋊ C₄⋊C₈
Sylow	C₅ ⋊ C₃×C₄⋊C₈

Subgroups

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

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11746488, #11746490, #11746492, …
maximal normal subgroup of	#11746488, #11746490, #11746492, …
maximal quotient of	C₃×C₄⋊#674, C₃×#64881, C₃×#64882, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 281);