C₁₀₉×#7490 #42125484

order	27904
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₉×#7490

Subgroups

	subgroup	G/H
center	C₄×C₁₀₉	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₄²×C₁₀₉
Frattini, cosocle	#532	C₂²×C₁₀₉
socle	C₂×C₁₀₉	#1390
prime radical	#532×C₁₀₉	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ #532×C₁₀₉ ◁ G

274 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Conjugacy classes & character table

3706 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 478);