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C₂²⋊SL(2,𝔽₃) #996

order	96
automorphism	Aut(G) = #11921467 (order 1152) · Inn(G) = C₂²⋊A₄ · Out(G) = S₄
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂³

Aliases

unambiguous semidirect product

C₂²⋊SL(2,𝔽₃)

Splittings

Fitting, derived, Sylow

C₂²×Q₈ ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₂²⋊A₄
commutator, Fitting, cosocle, hypocenter	C₂²×Q₈	C₃
socle	C₂³	A₄
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂

41 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 12 points:

(2,3,4)(6,7,9)(10,12,11);(5,9,8,11)(6,12,10,7);(1,4)(2,3)

Polycyclic presentation:

3,2^5|d^2=f,e^2=f,b^a=c,c^a=bc,d^a=e,e^a=de,e^d=ef

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	1	3	3	6	6	6	6	16	16	16	16
order	1	2	2	2	4	4	4	4	3	3	6	6
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₃	1	1	1	1	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₄	2	-2	2	-2	0	0	0	0	-1	-1	1	1
χ₅	2	-2	2	-2	0	0	0	0	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₆	2	-2	2	-2	0	0	0	0	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₇	3	3	3	3	-1	-1	-1	-1	0	0	0	0
χ₈	3	3	-1	-1	3	-1	-1	-1	0	0	0	0
χ₉	3	3	-1	-1	-1	3	-1	-1	0	0	0	0
χ₁₀	3	3	-1	-1	-1	-1	3	-1	0	0	0	0
χ₁₁	3	3	-1	-1	-1	-1	-1	3	0	0	0	0
χ₁₂	6	-6	-2	2	0	0	0	0	0	0	0	0

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

Relationships

derived subgroup of	#5930, #5931, #68001, …
maximal subgroup of	#393⋊C₃, #395⋊C₃, #5930, …
maximal normal subgroup of	#393⋊C₃, #395⋊C₃, #5930, …
maximal quotient of	#543⋊C₃, Q₈⋊SL(2,𝔽₃), C₂×C₂²⋊SL(2,𝔽₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 203);