C₁₀₉×#7155 #42125149

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₉×#7155

Subgroups

	subgroup	G/H
center	C₄²×C₁₀₉	D₈
commutator	C₈	C₄×C₈×C₁₀₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₀₉
socle	C₂²×C₁₀₉	#340
prime radical	C₂³×C₁₀₉	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

222 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 13952

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

(1,2,3,5,4,6,7,8)(9,10)(11,20)(12,25)(13,28)(14,30)(15,19)(16,21)(17,27)(18,26)(22,32)(23,40)(24,36)(29,33)(31,39)(34,38)(35,37)(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);(9,11,13,22,26,37,27,38,15,24,12,21,14,23,29,39)(10,16,17,20,30,34,28,40,19,32,33,36,18,31,25,35)

Conjugacy classes & character table

9592 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 143);