Finite Group Databasebeta

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C₂²×C₃ #24

order	12
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×D₃ (order 12) · Inn(G) = C₁ · Out(G) = C₂×D₃
holomorph	G ⋊ Aut(G) = D₃×S₄
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂

Aliases

abelian	C₂²×C₃
multiplicative	(ℤ/21)ˣ, (ℤ/28)ˣ, (ℤ/36)ˣ, (ℤ/42)ˣ

Subgroups

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

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 7 points:

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

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	3	3	2	6	6	2	6	6	2	6	6
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	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	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	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	1	1	1	-1	-1	-1
χ₅	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	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	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₇	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1
χ₈	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₉	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₁₀	1	1	1	-1	-1	-1	-1	-1	-1	1	1	1
χ₁₁	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₁₂	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	-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×C₂²⋊C₄, C₃×C₄⋊C₄, C₂×C₃×D₄, …
derived subgroup of	D₃×A₄, #806, #823, …
Frattini subgroup of	C₄²⋊C₉, C₄×Dic₉, Dic₉⋊C₄, …
Fitting subgroup of	C₂×Dic₃, #67, C₂²×D₃, …
socle of	C₂×Dic₃, C₂×C₃×C₄, C₂²×D₃, …
maximal subgroup of	C₂×Dic₃, #67, C₂×C₃×C₄, …
maximal normal subgroup of	C₂×Dic₃, #67, C₂×C₃×C₄, …
maximal quotient of	C₂×C₃×C₄, C₃×D₄, C₃×Q₈, …
automorphism group of	C₃×C₇, C₄×C₇, C₄×C₉, …
outer automorphism group of	C₃×C₇, C₄×C₇, C₄×C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12, 5);