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C₃×#157 #1109

order	108
isoclinism	isoclinic class of #157
automorphism	Aut(G) = C₂×AΓL(1,𝔽₉) (order 288) · Inn(G) = #157 · Out(G) = C₂³
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₃

Aliases

Splittings

Fitting, socle, Sylow	C₃³ ⋊ C₄
derived	C₃² ⋊ C₃×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#157
commutator, hypocenter	C₃²	C₃×C₄
Frattini	C₁	G
Fitting, socle	C₃³	C₄
prime radical	C₃×C₃⋊D₃	C₂
cosocle	C₃⋊D₃	C₂×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₃

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:

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

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#89091, #89092, AGL(1,𝔽₉)⋊D₃, …
maximal subgroup of	C₃×AGL(1,𝔽₉), C₃⋊AGL(1,𝔽₉), D₃×#157, …
maximal normal subgroup of	C₃×AGL(1,𝔽₉), C₃⋊AGL(1,𝔽₉), D₃×#157, …
maximal quotient of	C₃×#622, C₂×C₃×#157, C₉×#157, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(108, 36);