C₁₃×C₂₇⋊C₃ #11760412

order	1053
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₉×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×C₂₇⋊C₃
unambiguous semidirect product	C₂₇⋊(C₃×C₁₃)

Subgroups

	subgroup	G/H
center	C₉×C₁₃	C₃²
commutator	C₃	C₃×C₉×C₁₃
Frattini, cosocle	C₉	C₃²×C₁₃
socle	C₃×C₁₃	C₃×C₉
prime radical	C₃²×C₁₃	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₁₃ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 351

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,16,20,26,22,29,34,17,21,25,23,30,33,31,37,39,24,19,27,32,28,35,38,36,40);(15,19,21)(18,25,27)(20,28,30)(26,33,35)(29,36,37)(34,39,40)

Conjugacy classes & character table

429 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#15691491, #15691500, #15691503, …
maximal subgroup of	C₁₃×#6869, C₁₃×#6872, C₁₃×#6873, …
maximal normal subgroup of	C₁₃×#6869, C₁₃×#6872, C₁₃×#6873, …
maximal quotient of	C₁₃×#6864, C₁₃×C₃²⋊C₂₇, C₁₃×C₉⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1053, 6);

C13×C27⋊C3 #11760412