C₁₀₉×#1828 #28359082

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#1828

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#381
commutator	C₂³	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂⁴	C₂³×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂⁴×C₁₀₉ ◁ G

296 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 125
transitive degree	μ_t(G) = 1744

A minimal faithful permutation representation — 3 generators on 125 points:

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

Conjugacy classes & character table

3161 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#12725, C₁₀₉×#12726, C₁₀₉×#12727, …
maximal normal subgroup of	C₁₀₉×#12725, C₁₀₉×#12726, C₁₀₉×#12727, …
maximal quotient of	C₁₀₉×#8734, C₁₀₉×#8735, C₁₀₉×#8736, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 560);