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C₄²×AGL(1,𝔽₁₃) #72229079

order	2496
isoclinism	isoclinic class of AGL(1,𝔽₁₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₄³ · Schur multiplier (H₂) = C₄³

Aliases

C₄²×AGL(1,𝔽₁₃)

Splittings

Fitting	C₄²×C₁₃ ⋊ C₃×C₄
derived, Sylow	C₁₃ ⋊ C₃×C₄³
Hall	C₄²×#261 ⋊ C₃
Hall	C₁₃⋊C₃ ⋊ C₄³

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	AGL(1,𝔽₁₃)
commutator, hypocenter	C₁₃	C₃×C₄³
Frattini	C₂²	C₂²×AGL(1,𝔽₁₃)
Fitting	C₄²×C₁₃	C₃×C₄
socle	C₂²×C₁₃	C₂²×C₃×C₄
prime radical	C₂²×D₁₃⋊C₃	C₂³
cosocle	C₂²×D₁₃	C₂³×C₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₄²

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 208

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#127×AGL(1,𝔽₁₃), #104×AGL(1,𝔽₁₃), #97×AGL(1,𝔽₁₃)
maximal normal subgroup of	#127×AGL(1,𝔽₁₃), #104×AGL(1,𝔽₁₃), #97×AGL(1,𝔽₁₃)
maximal quotient of	#97×AGL(1,𝔽₁₃)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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