C₁₀₉×#1788 #28359042

order	13952
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₀₉×#1788

Subgroups

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

612 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 6976

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

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

Conjugacy classes & character table

4142 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#8465, C₁₀₉×#8467, C₁₀₉×#9345, …
maximal normal subgroup of	C₁₀₉×#8465, C₁₀₉×#8467, C₁₀₉×#9345, …
maximal quotient of	C₁₀₉×#8312, C₁₀₉×#8315, C₁₀₉×#8330, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 520);