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#126⋊C₃ #864

order	96
isoclinism	isoclinic class of #126⋊C₃
automorphism	Aut(G) = #560⋊C₃ (order 192) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

Splittings

Fitting, Sylow	#126 ⋊ C₃
derived	C₄² ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₄²	C₂×C₃
Frattini, socle	C₂²	C₂×A₄
Fitting	#126	C₃
LCS	G ▷ C₄²
UCS	C₁

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 2 generators on 16 points:

(2,10,11,6,3,13)(4,5,7)(8,9,12,16,14,15);(1,13,5,16)(2,10,9,3)(4,8,7,11)(6,15,14,12)

Polycyclic presentation:

2,3,2^4|c^2=ef,d^2=e,c^a=ce,d^a=df,c^b=d,d^b=cdf,e^b=f,f^b=ef

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀
size	1	3	4	6	6	12	16	16	16	16
order	1	2	2	4	4	4	3	3	6	6
χ₁	1	1	1	1	1	1	1	1	1	1
χ₂	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	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₄	1	1	-1	1	1	-1	1	1	-1	-1
χ₅	1	1	-1	1	1	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₆	1	1	-1	1	1	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₇	3	3	3	-1	-1	-1	0	0	0	0
χ₈	3	3	-3	-1	-1	1	0	0	0	0
χ₉	6	-2	0	$2 i$	$- 2 i$	0	0	0	0	0
χ₁₀	6	-2	0	$- 2 i$	$2 i$	0	0	0	0	0

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

Relationships

maximal subgroup of	C₂×#126⋊C₃, #560⋊C₃, C₃⋊(#126⋊C₃), …
maximal normal subgroup of	C₂×#126⋊C₃, #560⋊C₃, C₃×#126⋊C₃, …
maximal quotient of	#383⋊C₃, #396⋊C₃, #401⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 71);