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C₈×#170 #64878

order	320
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₈

Aliases

Splittings

C₅ ⋊ C₈²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₈²
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle, prime radical	C₂²×C₅	C₄²
cosocle	C₄×Dic₅	C₂²
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₈

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 21 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598623, C₁₆×#170, #10598636, …
maximal normal subgroup of	#10598623, C₁₆×#170, #10598636, …
maximal quotient of	C₈×#674, #10598512, C₁₆×#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 216);