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C₄×C₉×H(3,𝔽₃) #11757653

order	972
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄×C₉ · Schur multiplier (H₂) = C₃⁴

Aliases

C₄×C₉×H(3,𝔽₃)

Subgroups

	subgroup	G/H
center	C₃×C₄×C₉	C₃²
commutator	C₃	C₃²×C₄×C₉
Frattini, socle, cosocle	C₂×C₃²	C₂×C₃³
prime radical	C₂×C₃×H(3,𝔽₃)	C₂×C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₉ ◁ G

303 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 324

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13);(14,15,17)(16,19,21)(18,22,20);(15,18,19)(17,21,22)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₉×(C₃⋊(C₃⋊C₈))⋊C₃, H(3,𝔽₃)×C₉⋊C₈, #14516330, …
maximal subgroup of	C₉×(C₃⋊(C₃⋊C₈))⋊C₃, H(3,𝔽₃)×C₉⋊C₈, #14516330, …
maximal normal subgroup of	C₉×(C₃⋊(C₃⋊C₈))⋊C₃, H(3,𝔽₃)×C₉⋊C₈, #14516330, …
maximal quotient of	C₈×C₉×H(3,𝔽₃), C₂×C₄×C₉×H(3,𝔽₃)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(972, 378);