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C₃×C₄ #21

order	12
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂² (order 4) · Inn(G) = C₁ · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = D₃×D₄
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₂
abelian	C₃×C₄
multiplicative	(ℤ/13)ˣ, (ℤ/26)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃
socle, prime radical	C₂×C₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 7
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 1 generator on 7 points:

(1,2,3,4)(5,6,7)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	1	1	1	1	1	1	1	1	1	1	1
order	1	4	2	4	3	12	6	12	3	12	6	12
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	$i$	-1	$- i$	1	$i$	-1	$- i$	1	$i$	-1	$- i$
χ₃	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1
χ₄	1	$- i$	-1	$i$	1	$- i$	-1	$i$	1	$- i$	-1	$i$
χ₅	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₆	1	$i$	-1	$- i$	$\frac{- 1 + i \sqrt{3}}{2}$	ζ⁷	$\frac{1 - i \sqrt{3}}{2}$	−ζ⁷	$\frac{- 1 - i \sqrt{3}}{2}$	ζ¹¹	$\frac{1 + i \sqrt{3}}{2}$	−ζ¹¹
χ₇	1	-1	1	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₈	1	$- i$	-1	$i$	$\frac{- 1 + i \sqrt{3}}{2}$	−ζ⁷	$\frac{1 - i \sqrt{3}}{2}$	ζ⁷	$\frac{- 1 - i \sqrt{3}}{2}$	−ζ¹¹	$\frac{1 + i \sqrt{3}}{2}$	ζ¹¹
χ₉	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₁₀	1	$i$	-1	$- i$	$\frac{- 1 - i \sqrt{3}}{2}$	ζ¹¹	$\frac{1 + i \sqrt{3}}{2}$	−ζ¹¹	$\frac{- 1 + i \sqrt{3}}{2}$	ζ⁷	$\frac{1 - i \sqrt{3}}{2}$	−ζ⁷
χ₁₁	1	-1	1	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₁₂	1	$- i$	-1	$i$	$\frac{- 1 - i \sqrt{3}}{2}$	−ζ¹¹	$\frac{1 + i \sqrt{3}}{2}$	ζ¹¹	$\frac{- 1 + i \sqrt{3}}{2}$	−ζ⁷	$\frac{1 - i \sqrt{3}}{2}$	ζ⁷

Here ζ is a primitive root of unity of order 12 (ζ¹² = 1); i = ζ³.

Relationships

center of	C₃×M₁₆, C₃×#41, C₃×C₃⋊C₈, …
derived subgroup of	#204, D₂₄, Dic₁₂, …
Frattini subgroup of	C₉⋊C₈, C₈×C₉, Dic₃₆, …
Fitting subgroup of	C₃⋊C₈, Dic₆, C₄×D₃, …
maximal subgroup of	C₃⋊C₈, C₃×C₈, Dic₆, …
maximal normal subgroup of	C₃⋊C₈, C₃×C₈, Dic₆, …
maximal quotient of	C₃×C₈, C₂×C₃×C₄, C₄×C₉, …
automorphism group of	C₁₃, C₂×C₁₃
outer automorphism group of	C₁₃, C₂×C₁₃, C₅×D₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12, 2);