C₁₀₃×#1521 #27190887

order	13184
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₃×#1521

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₀₃
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₃
prime radical	#121×C₁₀₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₃ ◁ C₂×C₄²×C₁₀₃ ◁ G

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

normal subgroups only

This group has 218 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) = 135
transitive degree	μ_t(G) = 3296

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

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,32)(29,31)(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)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19)(18,22)(20,29)(21,26)(23,31)(24,32)(25,28)(27,30);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,23)(19,26)(24,28)(27,29)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₃×#8500, C₁₀₃×#8501, C₁₀₃×#8502, …
maximal normal subgroup of	C₁₀₃×#8500, C₁₀₃×#8501, C₁₀₃×#8502, …
maximal quotient of	C₁₀₃×#7580, C₁₀₃×#7581, C₁₀₃×#7584, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 253);