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C₂×C₄×#170 #65746

order	320
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₂×C₄×#170

Splittings

C₅ ⋊ C₂×C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₄×C₈
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle, prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₄

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 19 points:

(8,9,10,11)(12,13,14,16,15,17,18,19);(3,4,5,6)(7,8,9,11,10);(1,2)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598721, #10598765, #10598787, …
maximal normal subgroup of	#10598721, #10598765, #10598787, …
maximal quotient of	C₂×C₄×#674, C₂×#64859, C₄×#3998, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1084);