C₁₀₉×#12536 #42130530

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

Aliases

direct product

C₁₀₉×#12536

Subgroups

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

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

normal subgroups only

This group has 260 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) = 141
transitive degree	μ_t(G) = 13952

A minimal faithful permutation representation — 3 generators on 141 points:

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

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, 5524);