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C₂×C₈×AGL(1,𝔽₅) #65716

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

Aliases

C₂×C₈×AGL(1,𝔽₅)

Splittings

Fitting	C₂×C₅×C₈ ⋊ C₄
derived, Sylow	C₅ ⋊ C₂×C₄×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂×C₄×C₈
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂²×C₅	C₄²
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₄×D₅	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₈

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	(C₂×AGL(1,𝔽₅))⋊C₁₆, #10598670, C₄×C₈×AGL(1,𝔽₅), …
maximal normal subgroup of	(C₂×AGL(1,𝔽₅))⋊C₁₆, #10598670, C₄×C₈×AGL(1,𝔽₅), …
maximal quotient of	C₈×D₅⋊C₈, #10603806, C₈×#700, …
automorphism group of	#3926, C₃×#3928

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1054);