C₃²×H(5,𝔽₃) #14555446

order	2187
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃⁶ · Schur multiplier (H₂) = C₃¹⁴

Aliases

direct product

C₃²×H(5,𝔽₃)

Subgroups

	subgroup	G/H
center, socle	C₃³	C₃⁴
commutator, Frattini, cosocle	C₃	C₃⁶
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃ ▷ C₁

66734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 6 generators on 33 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×3₋¹⁺⁶, C₃×H(7,𝔽₃), #72264875, …
maximal normal subgroup of	C₃×3₋¹⁺⁶, C₃×H(7,𝔽₃), #72264875, …
maximal quotient of	#72264807

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 9305);

C32×H(5,𝔽3) #14555446