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C₂×#638 #3834

order	144
isoclinism	isoclinic class of #634
automorphism	Aut(G) = #120×AGL(2,𝔽₃) (order 13824) · Inn(G) = C₂×C₃⋊D₃ · Out(G) = C₂×#5926
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃

Aliases

Splittings

Fitting	C₂³×C₃² ⋊ C₂
Sylow	C₃² ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂³
Frattini	C₂	C₂²×C₃⋊D₃
Fitting	C₂³×C₃²	C₂
socle	C₂²×C₃²	C₂²
hypercenter	C₂³	C₃⋊D₃
hypocenter	C₃²	C₂×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 12 points:

(3,4,5)(10,12);(1,2)(4,5)(7,8)(9,10)(11,12);(1,2)(6,7,8)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11798540, #11798541, #11798586, …
maximal subgroup of	#63581, #63633, #63786, …
maximal normal subgroup of	#63581, #63633, #63786, …
maximal quotient of	C₂×#3750, #64133, C₂×#3752, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 177);