C₁₀₉×#7402 #42125396

order	27904
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#7402

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#353
commutator	C₂×C₄²	C₂×C₄×C₁₀₉
Frattini, cosocle	#390	C₂²×C₁₀₉
prime radical	C₁₀₉×#2011	C₂
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂³×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ #390×C₁₀₉ ◁ G

334 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 133
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 2 generators on 133 points:

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

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, 390);