C₁₀₉×#1763 #28359017

order	13952
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₀₉×#1763

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×D₄
commutator	C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

324 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 13952

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

(1,2)(3,4)(5,6,8,7)(9,10)(11,12)(13,15)(14,16)(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);(2,4)(9,11,13,16,14,15,12,10);(5,7,8,6)(9,12,14,13)(10,11,16,15)

Conjugacy classes & character table

4796 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#8417, C₁₀₉×#8419, C₁₀₉×#9164, …
maximal normal subgroup of	C₁₀₉×#8417, C₁₀₉×#8419, C₁₀₉×#9164, …
maximal quotient of	C₁₀₉×#8171, C₁₀₉×#8173, C₁₀₉×#8178, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 495);