C₁₀₉×#12771 #42130765

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₀₉×#12771

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#1790
commutator	C₂×D₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×#41	C₂³×C₁₀₉
prime radical	#532×C₁₀₉	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ C₂×#41×C₁₀₉ ◁ G

728 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 141
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 3 generators on 141 points:

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

Conjugacy classes & character table

3706 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27904, 5759);