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D₅×C₄⋊Q₈ #66057

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

Aliases

D₅×C₄⋊Q₈

Splittings

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

Subgroups

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

290 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);(10,12,13,17)(11,14,15,16);(7,9)(10,11,13,15)(12,16,17,14);(2,3)(4,5)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#332, #10599213, #10599215, …
maximal normal subgroup of	D₅×#332, #10599213, #10599215, …
maximal quotient of	#10606051, #10606057, #10606060, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1395);