C₁₀₉×#11769 #42129763

order	27904
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₉×#11769

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#1505
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₀₉
prime radical	#469×C₁₀₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ #405×C₁₀₉ ◁ G

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

normal subgroups only

This group has 304 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) = 141
transitive degree	μ_t(G) = 3488

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

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

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, 4757);