C₄²×C₂₇⋊C₃ #13046844

order	1296
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₄²×C₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₄²×C₂₇⋊C₃

Subgroups

	subgroup	G/H
center	C₄²×C₉	C₃²
commutator	C₃	C₃×C₄²×C₉
Frattini, cosocle	C₂²×C₉	C₂²×C₃²
socle	C₂²×C₃	C₂²×C₃×C₉
prime radical	C₂²×C₃²	C₂²×C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₄²×C₉ ◁ G

180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2,3,4)(9,10,13,11,15,21,17,24,29,12,16,20,18,25,28,26,32,34,19,14,22,27,23,30,33,31,35);(5,6,7,8)(10,14,16)(13,20,22)(15,23,25)(21,28,30)(24,31,32)(29,34,35)

Conjugacy classes & character table

528 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₄²×D₂₇⋊C₃, #72276094
maximal subgroup of	C₄²×D₂₇⋊C₃, #127×C₂₇⋊C₃, #72276094
maximal normal subgroup of	C₄²×D₂₇⋊C₃, #127×C₂₇⋊C₃, #72276094

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1296, 562);

C42×C27⋊C3 #13046844

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₄²×C₂₇⋊C₃ #13046844