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C₄×#674 #64857

order	320
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₄

Aliases

Splittings

C₅ ⋊ C₄×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₁₆
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₄²×C₅	C₄
socle, prime radical	C₂²×C₅	C₂×C₈
cosocle	C₂×#168	C₂²
LCS	G ▷ C₅
UCS	C₁ ◁ C₄²

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 25 points:

(2,3,4,5)(10,11,12,15,13,16,18,21,14,17,19,22,20,23,24,25);(1,2,3,5,4)(6,7,8,9)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×#674, #10598512, #10598513, …
maximal normal subgroup of	C₈×#674, #10598512, #10598513, …
maximal quotient of	C₈×#674, #10598513, C₄×#3928, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 195);