C₁₀₉×#10295 #42128289

order	27904
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#10295

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

504 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 141
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 141 points:

(1,2,5,9)(3,7,11,16)(4,6,12,8)(10,15,13,14)(17,18)(19,22)(20,23)(21,25)(24,29)(26,30)(27,32)(28,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(2,6)(4,12)(7,15)(8,9)(10,13)(14,16)(17,19,21,27)(18,22,25,32)(20,28,29,26)(23,31,24,30);(1,3,5,11)(2,7,9,16)(4,13,12,10)(6,14,8,15)(18,23)(20,29)(22,30)(24,25)(26,28)(31,32)

Conjugacy classes & character table

4360 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 3283);