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#120×AGL(1,𝔽₁₃) #72229048

order	4992
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

#120×AGL(1,𝔽₁₃)

Splittings

Fitting	C₁₃×#120 ⋊ C₃×C₄
Hall	#120×#261 ⋊ C₃
Hall	C₁₃⋊C₃ ⋊ C₄×#120
Sylow	C₁₃ ⋊ C₃×C₄×#120

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₁₃)
commutator, socle	C₂²×C₁₃	C₂³×C₃×C₄
Fitting	C₁₃×#120	C₃×C₄
prime radical	#120×D₁₃⋊C₃	C₂
cosocle	C₂²×D₁₃	C₂⁴×C₃
hypercenter	#120	AGL(1,𝔽₁₃)
hypocenter	C₁₃	C₃×C₄×#120
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #120

2448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	GL(2,ℤ/4)×AGL(1,𝔽₁₃), #1020×AGL(1,𝔽₁₃), B(4,𝔽₂)×AGL(1,𝔽₁₃)
maximal normal subgroup of	B(4,𝔽₂)×AGL(1,𝔽₁₃)
maximal quotient of	B(4,𝔽₂)×AGL(1,𝔽₁₃), #453×AGL(1,𝔽₁₃)
automorphism group of	C₄×Dic₁₃, C₄⋊Dic₁₃, C₂²⋊Dic₁₃, …

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