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C₂²×C₈×D₅ #66070

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×C₈×D₅

Splittings

Fitting	C₂²×C₅×C₈ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂³×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₅
commutator, hypocenter	C₅	C₂³×C₈
Frattini	C₄	C₂³×D₅
Fitting	C₂²×C₅×C₈	C₂
socle	C₂³×C₅	C₂×C₄
prime radical	C₂³×D₅	C₄
cosocle	C₄×C₅	C₂⁴
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₈

338 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)(3,4)(6,7)(8,9);(5,6,8,9,7)(10,11,12,14,13,15,16,17);(3,4);(1,2)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599440, #10599441, D₅×#336, …
maximal normal subgroup of	#10599440, #10599441, D₅×#336, …
maximal quotient of	C₂×C₈×Dic₁₀, C₈×#709, C₂×C₄×C₈×D₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1408);