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C₂²×C₄×Dic₅ #66116

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₂²×C₄×Dic₅

Splittings

C₅ ⋊ C₂²×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₅
commutator, hypocenter	C₅	C₂²×C₄²
Frattini	C₂²	C₂³×D₅
Fitting	C₂³×C₄×C₅	C₂
socle, prime radical	C₂⁴×C₅	C₂²
cosocle	C₂²×C₅	C₂⁴
LCS	G ▷ C₅
UCS	C₁ ◁ C₂³×C₄

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 320

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₅×#95, #10600194, #10600195, …
maximal normal subgroup of	Dic₅×#95, #10600194, #10600195, …
maximal quotient of	C₂²×C₄×#168, C₂²×#3935, C₂×C₄×#681, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1454);