C₁₀₇×AGL(1,ℤ/16) #28332864

order	13696
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₇×AGL(1,ℤ/16)

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₇	#442
commutator	C₈	C₂²×C₄×C₁₀₇
Frattini, cosocle	C₂×C₈	C₂³×C₁₀₇
prime radical	#509×C₁₀₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₇ ◁ C₂×C₄×C₁₀₇ ◁ #97×C₁₀₇ ◁ G

154 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 1712

A minimal faithful permutation representation — 3 generators on 123 points:

(2,6,8,15)(3,10)(4,11)(7,13,16,9)(12,14)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,2)(3,7)(4,13)(5,8)(6,14)(9,11)(10,15)(12,16);(1,3,11,14,5,12,4,10)(2,7,9,6,8,16,13,15)

Conjugacy classes & character table

2461 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₀₇×AGL(1,ℤ/16), C₁₀₇×#33468, C₁₀₇×#33469, …
maximal normal subgroup of	C₂×C₁₀₇×AGL(1,ℤ/16), C₁₀₇×#33468, C₁₀₇×#33469, …
maximal quotient of	C₁₀₇×#12065, C₁₀₇×#12067, C₁₀₇×#12077, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13696, 913);