C₁₀₉×#7493 #42125487

order	27904
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#7493

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#1390
commutator	C₂×D₄	C₄²×C₁₀₉
Frattini, cosocle	C₂×#127	C₂²×C₁₀₉
prime radical	C₂×#127×C₁₀₉	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂³×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×#127×C₁₀₉ ◁ G

334 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 125
transitive degree	μ_t(G) = 1744

A minimal faithful permutation representation — 2 generators on 125 points:

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

Conjugacy classes & character table

3052 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 481);