C₁₀₉×#1899 #28359153

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₁₀₉×#1899

Subgroups

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

378 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 129
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 3 generators on 129 points:

(1,2)(3,4)(5,6)(7,10,14,17)(8,11)(9,12)(13,16,19,20)(15,18)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(5,7)(6,10,12,17)(8,13)(9,14)(11,16,18,20)(15,19);(2,4)(5,8,9,15)(6,11,12,18)(7,13,14,19)(10,16,17,20)

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#8487, C₁₀₉×#8488, C₂×C₁₀₉×#1899, …
maximal normal subgroup of	C₁₀₉×#8487, C₁₀₉×#8488, C₂×C₁₀₉×#1899, …
maximal quotient of	C₁₀₉×#9080, C₁₀₉×#9082, C₁₀₉×#9084, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 631);