AGL(1,𝔽₅)×C₁₀₉ #14546125

order	2180
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₀₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

AGL(1,𝔽₅)×C₁₀₉

Splittings

Fitting, socle, Hall	C₅×C₁₀₉ ⋊ C₄
derived, Sylow	C₅ ⋊ C₄×C₁₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₀₉	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₁₀₉
Frattini	C₁	G
Fitting, socle	C₅×C₁₀₉	C₄
prime radical	D₅×C₁₀₉	C₂
cosocle	D₅	C₂×C₁₀₉
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₀₉

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 114
transitive degree	μ_t(G) = 545

A minimal faithful permutation representation — 2 generators on 114 points:

(2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114);(1,2,3,5,4)

Conjugacy classes & character table

545 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#1035×C₁₀₉, C₅×AGL(1,𝔽₅)×C₁₀₉, #1043×C₁₀₉, …
maximal normal subgroup of	C₅×AGL(1,𝔽₅)×C₁₀₉, C₇×AGL(1,𝔽₅)×C₁₀₉, C₁₁×AGL(1,𝔽₅)×C₁₀₉, …
maximal quotient of	#1035×C₁₀₉, C₅×AGL(1,𝔽₅)×C₁₀₉, #1043×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2180, 7);

AGL(1,𝔽5)×C109 #14546125