C₁₀₉×#12533 #42130527

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₀₉×#12533

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₀₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

238 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 27904

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

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,27)(20,23)(21,34)(22,35)(24,30)(25,28)(26,33)(29,39)(31,46)(32,47)(36,44)(37,43)(38,48)(40,42)(41,45)(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);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(18,23)(24,36)(25,37)(26,35)(27,39)(34,47)(38,40)(41,46);(3,10)(6,15)(9,16)(11,14)(17,19,28,42,22,31,32,44,20,29,43,48,33,45,21,30)(18,24,34,41,26,38,37,39,23,36,47,46,35,40,25,27)

Conjugacy classes & character table

5014 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 5521);