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C₂²×C₂²⋊Dic₅ #66173

order	320
isoclinism	isoclinic class of Dic₁₀
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

C₂²×C₂²⋊Dic₅

Splittings

C₅ ⋊ C₂²×C₂²⋊C₄

Subgroups

	subgroup	G/H
center	C₂⁴	C₂×D₅
commutator	C₂×C₅	C₂³×C₄
Frattini	C₂²	C₂³×D₅
Fitting, prime radical	C₂⁵×C₅	C₂
socle	C₂⁴×C₅	C₂²
cosocle	C₂²×C₅	C₂⁴
hypercenter	C₂⁵	D₅
hypocenter	C₅	C₂²×C₂²⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵

674 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:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600288, #10600289, C₂×#64746, …
maximal normal subgroup of	#10600288, #10600289, C₂×#64746, …
maximal quotient of	C₂²×C₂²⋊#168, C₂×#65496, C₂²×#3963, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1511);