C₁₀₉×#12723 #42130717

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

Aliases

direct product

C₁₀₉×#12723

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×#41	C₂³×C₁₀₉
prime radical	#535×C₁₀₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ C₄×#41×C₁₀₉ ◁ G

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

normal subgroups only

This group has 694 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)(3,12,15,27)(4,9,17,25)(5,10)(6,11)(7,13)(8,16,24,31)(14,20,18,32)(19,23)(21,28)(22,29)(26,30)(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);(1,3,6,15)(2,7,11,22)(4,13,17,29)(5,18,19,14)(8,20,24,32)(9,12,25,27)(10,26,23,21)(16,30,31,28);(1,4,6,17)(2,8)(3,13,15,29)(5,16,19,31)(7,20)(9,10)(11,24)(12,26)(14,28,18,30)(21,27)(22,32)(23,25)

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, 5711);