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Q₈×D₂₀ #65909

order	320
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

Q₈×D₂₀

Splittings

Fitting	C₄×C₅×Q₈ ⋊ C₂
Sylow	C₅ ⋊ D₄×Q₈

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₄×C₅×Q₈	C₂
prime radical	C₂×D₂₀	C₂²
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	D₄×Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 160

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

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599044, #10599059, #10599062, …
maximal normal subgroup of	#10599044, #10599059, #10599062, …
maximal quotient of	#10605381, #10605406, #10605413, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1247);