C₁₀₉×#7353 #42125347

order	27904
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₉×#7353

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#353
commutator	C₄²	C₂×C₈×C₁₀₉
Frattini, cosocle	#382	C₂²×C₁₀₉
prime radical	C₂³×C₁₀₉	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂³×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ #382×C₁₀₉ ◁ G

204 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 149
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 2 generators on 149 points:

(1,2,3,5,4,6,7,8)(9,10)(11,19,21,34)(12,23,24,39)(13,26,28,17)(14,18)(15,20,29,36)(16,25)(22,32,27,35)(30,37,31,38)(33,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149);(9,11,14,22)(10,15,18,31)(12,20,24,38)(13,27,28,21)(16,32,33,19)(17,29,26,30)(23,35,39,34)(25,36,40,37)

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, 341);