Finite Group Databasebeta

← Prev (#88746) Next (#88748) →

Q₈×#272 #88747

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

Aliases

Splittings

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

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂²×C₃⋊D₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³
Fitting	Q₈×H(3,𝔽₃)	C₂
prime radical	C₂×#272	C₂²
hypercenter	C₃×Q₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×Q₈
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈

209 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:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11720790, #11721226, #11721228, …
maximal normal subgroup of	#11720790, #11721226, #11721228, …
maximal quotient of	#11721755, #11721756, #11721757, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 394);