Finite Group Databasebeta

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C₃×A₄ #159

order	36
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = D₃×S₄ (order 144) · Inn(G) = A₄ · Out(G) = C₂×D₃
holomorph	G ⋊ Aut(G) = #72213328
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

Borel	B(2,𝔽₄)
direct product	C₃×A₄
unambiguous semidirect product	C₂²⋊C₃²
unambiguous semidirect product	(C₂²×C₃)⋊C₃

Splittings

Fitting, socle	C₂²×C₃ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃	A₄
commutator, cosocle, hypocenter	C₂²	C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₃

14 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:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	3	4	4	1	3	4	4	1	3	4	4
order	1	2	3	3	3	6	3	3	3	6	3	3
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	3	-1	0	0	3	-1	0	0	3	-1	0	0
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	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$
χ₇	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1
χ₈	3	-1	0	0	$\frac{- 3 + 3 i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	0	0	$\frac{- 3 - 3 i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	0	0
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	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	1
χ₁₁	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	$\frac{- 1 - i \sqrt{3}}{2}$
χ₁₂	3	-1	0	0	$\frac{- 3 - 3 i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	0	0	$\frac{- 3 + 3 i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	0	0

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

Relationships

derived subgroup of	C₃⋊S₄, C₃⋊SL(2,ℤ/4), D₃×S₄, …
maximal subgroup of	C₃×S₄, C₃⋊S₄, D₃×A₄, …
maximal normal subgroup of	C₃×S₄, C₃⋊S₄, D₃×A₄, …
maximal quotient of	C₃×SL(2,𝔽₃), C₂×C₃×A₄, C₉×A₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36, 11);