Finite Group Databasebeta

← Prev (#89078) Next (#89080) →

Q₈×C₃⋊(C₃⋊D₃) #89079

order	432
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃³

Aliases

Q₈×C₃⋊(C₃⋊D₃)

Splittings

Fitting	C₃³×Q₈ ⋊ C₂
Sylow	C₃³ ⋊ C₂×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂×C₃³	C₂³
Fitting	C₃³×Q₈	C₂
prime radical	C₂×C₃⋊(C₃⋊D₃)	C₂²
hypercenter	Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ Q₈

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11723571, #11723577, #11723588, …
maximal normal subgroup of	#11723571, #11723577, #11723588, …
maximal quotient of	#11723928, #11723929, #11723930, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 726);