Finite Group Databasebeta

← Prev (#3746) Next (#3748) →

C₂×C₃⋊(C₃⋊C₈) #3747

order	144
isoclinism	isoclinic class of C₃⋊D₃
automorphism	Aut(G) = C₂×D₄×AGL(2,𝔽₃) (order 6912) · Inn(G) = C₃⋊D₃ · Out(G) = C₂×D₄×S₄
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₂×C₃⋊(C₃⋊C₈)
unambiguous semidirect product	(C₂×C₃)⋊(C₃⋊C₈)

Splittings

C₃² ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	C₃⋊D₃
commutator, hypocenter	C₃²	C₂×C₈
Frattini	C₄	C₂×C₃⋊D₃
Fitting	C₂×C₃²×C₄	C₂
socle, prime radical	C₂²×C₃²	C₄
cosocle	C₃²×C₄	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₄

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11771725, #11771726, #11771742, …
maximal subgroup of	Dic₃×C₃⋊C₈, #63554, (C₂×D₃)⋊(C₃⋊C₈), …
maximal normal subgroup of	Dic₃×C₃⋊C₈, #63554, (C₂×D₃)⋊(C₃⋊C₈), …
maximal quotient of	C₄×C₃⋊(C₃⋊C₈), C₄⋊(C₃⋊(C₃⋊C₈)), C₂×C₃⋊(C₃⋊C₁₆), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 90);