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Q₁₆×Dic₅ #65472

order	320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

Q₁₆×Dic₅

Splittings

C₅ ⋊ C₄×Q₁₆

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₂×C₅×Q₁₆	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂×Q₁₆	D₅
hypocenter	C₅	C₄×Q₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 25 points:

(2,3)(4,5)(6,7,8,9);(1,2,4,5,3)(10,12,14,20)(11,15,17,24)(13,22,23,19)(16,25,21,18);(10,11,14,17)(12,18,20,25)(13,21,23,16)(15,22,24,19)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598807, #10598809, #10599999, …
maximal normal subgroup of	#10598807, #10598809, #10599999, …
maximal quotient of	Dic₅×#103, #10601017, #10601029, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 810);