C₁₀₉×#7343 #42125337

order	27904
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#7343

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#1316
commutator	C₂²×C₄	C₂×C₈×C₁₀₉
Frattini, cosocle	#407	C₂²×C₁₀₉
prime radical	C₂⁴×C₁₀₉	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ #407×C₁₀₉ ◁ G

324 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 3488

A minimal faithful permutation representation — 2 generators on 141 points:

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

Conjugacy classes & character table

3052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 331);