Finite Group Databasebeta

← Prev (#88845) Next (#88847) →

C₃×#3755 #88846

order	432
isoclinism	isoclinic class of #3753
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₃

Aliases

Splittings

Fitting	C₃³×Q₈ ⋊ C₂
Sylow	C₃³ ⋊ SD₁₆

Subgroups

	subgroup	G/H
center	C₂×C₃	#638
commutator, cosocle	C₃²×C₄	C₂²×C₃
Frattini	C₄	C₂×C₃×C₃⋊D₃
Fitting	C₃³×Q₈	C₂
socle	C₂×C₃³	D₄
prime radical	C₃×C₄⋊(C₃⋊D₃)	C₂
hypercenter	C₃×Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×SD₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₈

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

81 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×#215, C₃×#63939, C₃×#63948, …
maximal normal subgroup of	C₃×D₃×#215, C₃×#63939, C₃×#63948, …
maximal quotient of	C₃×#63635, C₃×#63636, C₃×#63662, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 493);