C₁₀₉×#12338 #42130332

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

Aliases

direct product

C₁₀₉×#12338

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
prime radical	C₁₀₉×#3403	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ #446×C₁₀₉ ◁ G

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

normal subgroups only

This group has 320 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) = 149
transitive degree	μ_t(G) = 13952

A minimal faithful permutation representation — 3 generators on 149 points:

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

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, 5326);