C₁₀₉×#12530 #42130524

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₁₀₉×#12530

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

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

normal subgroups only

This group has 284 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) = 149
transitive degree	μ_t(G) = 13952

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

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

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, 5518);