C₂×#352×C₁₀₃ #27191485

order	13184
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#352×C₁₀₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₃	#99
commutator	C₂³	C₂²×C₄×C₁₀₃
Frattini, cosocle	C₂×D₄	C₂³×C₁₀₃
prime radical	C₂²×D₄×C₁₀₃	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₂³×C₁₀₃ ◁ C₂²×D₄×C₁₀₃ ◁ G

278 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 121
transitive degree	μ_t(G) = 3296

A minimal faithful permutation representation — 3 generators on 121 points:

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

Conjugacy classes & character table

2678 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×#352×C₁₀₃, C₁₀₃×#12621, C₁₀₃×#12688, …
maximal normal subgroup of	C₄×#352×C₁₀₃, C₁₀₃×#12621, C₁₀₃×#12688, …
maximal quotient of	C₂×C₁₀₃×#1317, C₂×C₁₀₃×#1326, C₁₀₃×#11472, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 851);