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C₃×C₃⋊(C₃⋊C₁₆) #88583

order	432
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₁₆ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×C₃⋊(C₃⋊C₁₆)
unambiguous semidirect product	C₃⋊(C₃×C₃⋊C₁₆)

Splittings

derived	C₃² ⋊ C₃×C₁₆
Sylow	C₃³ ⋊ C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₈	C₃⋊D₃
commutator, hypocenter	C₃²	C₃×C₁₆
Frattini	C₈	C₃×C₃⋊D₃
Fitting	C₃³×C₈	C₂
socle, prime radical	C₂×C₃³	C₈
cosocle	C₃²×C₈	C₂×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₃×C₈

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

(1,6,9,8,11,16,3,7,5,13,15,2,4,12,10,14)(17,18,19)(21,22)(24,25);(23,24,25);(20,21,22)

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13154954, #13154955, #13154956, …
maximal subgroup of	C₃×#63540, C₃⋊#63540, C₃×D₃×C₃⋊C₁₆, …
maximal normal subgroup of	C₃×#63540, C₃⋊#63540, C₃×D₃×C₃⋊C₁₆, …
maximal quotient of	C₃×C₃⋊(C₃⋊C₃₂), C₂×C₃×C₃⋊(C₃⋊C₁₆), C₃×C₃⋊(C₉⋊C₁₆), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 230);