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C₃⋊#157 #1110

order	108
isoclinism	isoclinic class of C₃⋊#157
automorphism	Aut(G) = D₃×AΓL(1,𝔽₉) (order 864) · Inn(G) = G · Out(G) = D₄
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

Aliases

unambiguous semidirect product

Splittings

Fitting, socle, derived, Sylow

C₃³ ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃³	C₄
prime radical, cosocle	C₃×C₃⋊D₃	C₂
LCS	G ▷ C₃³
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 9 points:

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

Polycyclic presentation:

2^2,3^3|a^2=b,c^a=c^2d,d^a=cd,e^a=e^2,c^b=c^2,d^b=d^2

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	2	4	4	4	4	4	4	9	18	27	27
order	1	3	3	3	3	3	3	3	2	6	4	4
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	1	1	-1	-1
χ₃	1	1	1	1	1	1	1	1	-1	-1	$i$	$- i$
χ₄	1	1	1	1	1	1	1	1	-1	-1	$- i$	$i$
χ₅	2	-1	2	2	-1	-1	-1	-1	2	-1	0	0
χ₆	2	-1	2	2	-1	-1	-1	-1	-2	1	0	0
χ₇	4	4	1	-2	1	1	-2	-2	0	0	0	0
χ₈	4	4	-2	1	-2	-2	1	1	0	0	0	0
χ₉	4	-2	1	-2	$\frac{- 1 - 3 i \sqrt{3}}{2}$	$\frac{- 1 + 3 i \sqrt{3}}{2}$	1	1	0	0	0	0
χ₁₀	4	-2	1	-2	$\frac{- 1 + 3 i \sqrt{3}}{2}$	$\frac{- 1 - 3 i \sqrt{3}}{2}$	1	1	0	0	0	0
χ₁₁	4	-2	-2	1	1	1	$\frac{- 1 - 3 i \sqrt{3}}{2}$	$\frac{- 1 + 3 i \sqrt{3}}{2}$	0	0	0	0
χ₁₂	4	-2	-2	1	1	1	$\frac{- 1 + 3 i \sqrt{3}}{2}$	$\frac{- 1 - 3 i \sqrt{3}}{2}$	0	0	0	0

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

Relationships

maximal subgroup of	D₃×#157, D₃²⋊D₃, C₃⋊PSU(3,𝔽₂), …
maximal normal subgroup of	D₃×#157, D₃²⋊D₃, C₃⋊PSU(3,𝔽₂), …
maximal quotient of	C₃⋊#622, C₂×C₃⋊#157, C₉⋊#157, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 37);