C₁₀₉×#12874 #42130868

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#12874

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#1823
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
prime radical	C₂×#41×C₁₀₉	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂×#131×C₁₀₉ ◁ G

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

normal subgroups only

This group has 376 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:

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

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, 5862);