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Q₈×(C₃⋊D₃)⋊C₃ #88721

order	432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₃

Aliases

Q₈×(C₃⋊D₃)⋊C₃

Splittings

Fitting	Q₈×H(3,𝔽₃) ⋊ C₂
Sylow	H(3,𝔽₃) ⋊ C₂×Q₈

Subgroups

	subgroup	G/H
center	C₂	C₂²×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂×C₃²	C₂³×C₃
Frattini, socle	C₂×C₃	C₂²×C₃×D₃
Fitting	Q₈×H(3,𝔽₃)	C₂
prime radical	C₂×(C₃⋊D₃)⋊C₃	C₂²
hypercenter	Q₈	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₂×C₃×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ Q₈

145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

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

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11721232, #11721234, #11721236, …
maximal normal subgroup of	#11721232, #11721234, #11721236, …
maximal quotient of	#64096⋊C₃, #64097⋊C₃, #64098⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 368);