C₁₀₉×#7160 #42125154

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#7160

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₀₉	D₈
commutator	C₈	C₄×C₈×C₁₀₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₀₉
socle	C₂²×C₁₀₉	#335
prime radical	C₂³×C₁₀₉	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

210 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 145
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 2 generators on 145 points:

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

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, 148);