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#127⋊C₃ #865

order	96
isoclinism	isoclinic class of #127⋊C₃
automorphism	Aut(G) = #4626⋊C₂ (order 384) · Inn(G) = G · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₄

Aliases

unambiguous semidirect product

Splittings

Fitting, Sylow	#127 ⋊ 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	#127	C₃
LCS	G ▷ C₄²
UCS	C₁

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 12 points:

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

Polycyclic presentation:

2,3,2^4|c^2=ef,d^2=e,c^a=cef,d^a=de,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	2	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	-2	0	0	0	0	0
χ₁₀	6	-2	0	-2	2	0	0	0	0	0

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

Relationships

derived subgroup of	#4626⋊C₂, (#560⋊C₃)⋊C₄, #4626⋊C₄, …
maximal subgroup of	#4626, #857⋊C₂, C₂×#127⋊C₃, …
maximal normal subgroup of	#4626, #857⋊C₂, C₂×#127⋊C₃, …
maximal quotient of	#384⋊C₃, #392⋊C₃, #399⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 72);