C₁₀₉×#12326 #42130320

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

Aliases

direct product

C₁₀₉×#12326

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
prime radical	C₁₀₉×#3273	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ #436×C₁₀₉ ◁ G

390 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 6976

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

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

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, 5314);