C₁₀₃×#1497 #27190863

order	13184
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₃×#1497

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₀₃
Frattini, cosocle	C₄²	C₂³×C₁₀₃
socle	C₂²×C₁₀₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₃ ◁ C₂×C₄²×C₁₀₃ ◁ G

202 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 135
transitive degree	μ_t(G) = 6592

A minimal faithful permutation representation — 3 generators on 135 points:

(1,2,4,8,5,7,11,14)(3,6,10,12,9,13,15,16)(17,18)(19,26)(20,23)(21,25)(22,27)(24,30)(28,31)(29,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,28)(18,22,25,32)(20,29,30,27)(23,26,24,31);(2,7)(3,9)(8,14)(10,15)(17,20,21,30)(18,23,25,24)(19,27,28,29)(22,31,32,26)

Conjugacy classes & character table

3296 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₃×#8490, C₁₀₃×#8501, C₁₀₃×#8502, …
maximal normal subgroup of	C₁₀₃×#8490, C₁₀₃×#8501, C₁₀₃×#8502, …
maximal quotient of	C₁₀₃×#7701, C₁₀₃×#7702, C₁₀₃×#7703, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 229);