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C₂×A₄ #72

order	24
isoclinism	isoclinic class of A₄
automorphism	Aut(G) = S₄ (order 24) · Inn(G) = A₄ · Out(G) = C₂
holomorph	G ⋊ Aut(G) = C₂×A₄⋊S₄
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

affine special semilinear	AΣL(1,𝔽₈)
direct product	C₂×A₄
unambiguous semidirect product	C₂³⋊C₃

Splittings

Fitting, socle, Sylow	C₂³ ⋊ C₃
derived	C₂² ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	A₄
commutator, cosocle, hypocenter	C₂²	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₂³	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 6 points:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈
size	1	3	4	4	1	3	4	4
order	1	2	3	3	2	2	6	6
χ₁	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	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	3	-1	0	0	3	-1	0	0
χ₅	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	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₇	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	-1	-1	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₈	3	-1	0	0	-3	1	0	0

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

Relationships

derived subgroup of	A₄⋊Q₈, C₄⋊S₄, GL(2,ℤ/4), …
maximal subgroup of	SL(2,ℤ/4), C₄×A₄, C₂×S₄, …
maximal normal subgroup of	SL(2,ℤ/4), C₄×A₄, C₂×S₄, …
maximal quotient of	C₄×A₄, C₂×SL(2,𝔽₃), #41⋊C₃, …
outer automorphism group of	#126, #2032

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24, 13);