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D₃×A₄ #647

order	72
isoclinism	isoclinic class of D₃×A₄
automorphism	Aut(G) = D₃×S₄ (order 144) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = D₃²×A₄⋊S₄
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×A₄
unambiguous semidirect product	C₃⋊(C₂×A₄)
unambiguous semidirect product	(C₂²×D₃)⋊C₃

Splittings

Fitting, socle, derived

C₂²×C₃ ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₃	C₂×C₃
LCS	G ▷ C₂²×C₃
UCS	C₁

26 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)(6,7);(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	2	6	8	8	3	9	12	12
order	1	2	3	3	3	6	3	3	2	2	6	6
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	1	1	1	1	-1	-1	-1	-1
χ₆	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	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₇	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	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₈	3	-1	0	0	3	-1	0	0	-3	1	0	0
χ₉	2	2	2	2	-1	-1	-1	-1	0	0	0	0
χ₁₀	2	2	$- 1 + i \sqrt{3}$	$- 1 - i \sqrt{3}$	-1	-1	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	0	0	0	0
χ₁₁	2	2	$- 1 - i \sqrt{3}$	$- 1 + i \sqrt{3}$	-1	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	0	0	0	0
χ₁₂	6	-2	0	0	-3	1	0	0	0	0	0	0

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

Relationships

maximal subgroup of	D₃×S₄, C₂×D₃×A₄, D₉⋊A₄, …
maximal normal subgroup of	D₃×S₄, C₂×D₃×A₄, C₃×D₃×A₄, …
maximal quotient of	C₃⋊(#41⋊C₃), D₃×SL(2,𝔽₃), Dic₃×A₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 44);