Finite Group Databasebeta

← Prev (#795) Next (#797) →

#95⋊C₃ #796

order	96
isoclinism	isoclinic class of #95⋊C₃
automorphism	Aut(G) = #4626⋊C₂ (order 384) · Inn(G) = C₄²⋊C₃ · Out(G) = D₄
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

#95⋊C₃

Splittings

Fitting, derived, Sylow

#95 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₄²⋊C₃
commutator, Fitting, cosocle, hypocenter	#95	C₃
Frattini, socle	C₂³	A₄
LCS	G ▷ #95
UCS	C₁ ◁ C₂

21 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)(3,7,12)(4,8,11)(6,10,9);(1,3,4,9)(5,10)(6,7)(11,12)

Polycyclic presentation:

3,2^5|b^2=de,c^2=d,b^a=c,c^a=bce,d^a=ef,e^a=def,c^b=cf

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	1	1	$- 1 + 2 i$	$- 1 - 2 i$	0	0	0	0
χ₉	3	3	-1	-1	1	1	$- 1 - 2 i$	$- 1 + 2 i$	0	0	0	0
χ₁₀	3	3	-1	-1	$- 1 + 2 i$	$- 1 - 2 i$	1	1	0	0	0	0
χ₁₁	3	3	-1	-1	$- 1 - 2 i$	$- 1 + 2 i$	1	1	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	#4621, #4622, SL(2,ℤ/8), …
maximal subgroup of	#4621, #4622, C₂×#95⋊C₃, …
maximal normal subgroup of	#4621, #4622, C₂×#95⋊C₃, …
maximal quotient of	#338⋊C₃, C₂×#95⋊C₃, #95⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 3);