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#120⋊C₃ #863

order	96
isoclinism	isoclinic class of #120⋊C₃
automorphism	Aut(G) = #4625⋊C₂ (order 384) · Inn(G) = G · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product

Splittings

Fitting, Sylow	#120 ⋊ C₃
derived	C₂⁴ ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂⁴	C₂×C₃
Frattini, socle	C₂²	C₂×A₄
Fitting	#120	C₃
LCS	G ▷ C₂⁴
UCS	C₁

35 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

(1,2)(3,6,7,5,4,8);(2,8)(5,6)

Polycyclic presentation:

2,3,2^4|c^a=ce,d^a=df,c^b=d,d^b=cd,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	2	2	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	-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	#4625⋊C₂, #5450⋊C₄, #4625⋊C₄, …
maximal subgroup of	#4625, #1020⋊C₂, C₂×#120⋊C₃, …
maximal normal subgroup of	#4625, #1020⋊C₂, C₂×#120⋊C₃, …
maximal quotient of	#379⋊C₃, #393⋊C₃, #395⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 70);