Finite Group Databasebeta

← Prev (#7) Next (#9) →

C₂×C₃ #8

order	6
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂ (order 2) · Inn(G) = C₁ · Out(G) = C₂
holomorph	G ⋊ Aut(G) = C₂×D₃
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₆
abelian	C₂×C₃
multiplicative	(ℤ/7)ˣ, (ℤ/9)ˣ, (ℤ/14)ˣ, (ℤ/18)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 5
transitive degree	μ_t(G) = 6

A minimal faithful permutation representation — 1 generator on 5 points:

(1,2)(3,4,5)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆
size	1	1	1	1	1	1
order	1	3	3	2	6	6
χ₁	1	1	1	1	1	1
χ₂	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	1	1	1	-1	-1	-1
χ₅	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₆	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$

Here i is the imaginary unit (i² = −1).

Relationships

center of	C₃×D₄, C₃×Q₈, C₃×Dic₃, …
derived subgroup of	Dic₆, D₁₂, #67, …
Frattini subgroup of	Dic₉, C₄×C₉, Q₈⋊C₉, …
Fitting subgroup of	Dic₃, C₂×D₃, C₃×SL(2,𝔽₅), …
socle of	Dic₃, C₃×C₄, C₂×D₃, …
maximal subgroup of	Dic₃, C₃×C₄, C₂×D₃, …
maximal normal subgroup of	Dic₃, C₃×C₄, C₂×D₃, …
maximal quotient of	C₃×C₄, C₂²×C₃, C₂×C₉, …
automorphism group of	C₇, C₉, C₂×C₇, …
outer automorphism group of	C₇, C₉, C₂×C₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6, 2);