AGL(1,ℤ/8)×C₁₀₉ #16812596

order	3488
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

AGL(1,ℤ/8)×C₁₀₉

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂³×C₁₀₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ G

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 872

A minimal faithful permutation representation — 3 generators on 117 points:

(1,2)(3,5)(4,6)(7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(2,5)(3,8)(6,7);(2,6)(5,7)

Conjugacy classes & character table

1199 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#28379030, #28379032, #28379069, …
maximal subgroup of	#453×C₁₀₉, #454×C₁₀₉, #471×C₁₀₉, …
maximal normal subgroup of	#453×C₁₀₉, #454×C₁₀₉, #471×C₁₀₉, …
maximal quotient of	#417×C₁₀₉, #418×C₁₀₉, #428×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3488, 43);

AGL(1,ℤ/8)×C109 #16812596