C₁₀₉×#12401 #42130395

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#12401

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
prime radical	C₂×D₄×C₁₀₉	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ #436×C₁₀₉ ◁ G

274 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 157
transitive degree	μ_t(G) = 13952

A minimal faithful permutation representation — 3 generators on 157 points:

(1,2,6,10)(3,11,14,15)(4,9,5,8)(7,13,16,12)(17,18,23,27)(19,28,31,46)(20,25,22,37)(21,33,36,44)(24,39,41,30)(26,32,45,35)(29,47,38,43)(34,40,48,42)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,19,23,31)(18,24,27,41)(20,29,22,38)(21,34,36,48)(25,40,37,42)(26,43,45,47)(28,44,46,33)(30,32,39,35);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,35,36,32)(24,40,41,42)(26,44,45,33)(28,43,46,47)(30,48,39,34)

Conjugacy classes & character table

3379 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 5389);