C₁₀₉×#12410 #42130404

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

Aliases

direct product

C₁₀₉×#12410

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₀₉	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈×Q₈×C₁₀₉ ◁ G

276 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 157
transitive degree	μ_t(G) = 27904

A minimal faithful permutation representation — 3 generators on 157 points:

(1,2,7,11)(3,12,4,15)(5,10,6,9)(8,14,16,13)(17,18,22,26)(19,27,30,46)(20,31,35,45)(21,36,33,25)(23,38,40,28)(24,41,43,34)(29,47,37,42)(32,44,48,39)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,10,12)(17,19,22,30)(18,23,26,40)(20,32,35,48)(21,37,33,29)(24,42,43,47)(25,44,36,39)(27,45,46,31)(28,41,38,34);(1,4,7,3)(2,8,11,16)(5,13,6,14)(9,15,10,12)

Conjugacy classes & character table

4360 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 5398);