C₁₀₉×#12706 #42130700

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

Aliases

direct product

C₁₀₉×#12706

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#1781
commutator	C₂⁴	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×D₄	C₂³×C₁₀₉
prime radical	C₁₀₉×#3485	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂³×C₁₀₉ ◁ C₂²×#41×C₁₀₉ ◁ G

1000 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,11)(4,8)(5,17,19,9)(6,10)(7,13,21,31)(12,27)(14,26)(15,24,18,22)(16,23)(20,28,32,30)(25,29)(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)(2,7)(4,12)(5,13)(6,14)(8,20)(9,11)(10,21)(15,28)(16,29)(17,26)(18,30)(19,31)(22,27)(23,32)(24,25);(1,4,6,16)(2,8,10,23)(3,12,14,29)(5,18,19,15)(7,20,21,32)(9,24,17,22)(11,25,26,27)(13,30,31,28)

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, 5694);