C₁₀₉×#3483 #28360737

order	13952
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₁₀₉ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₁₀₉×#3483

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
socle	C₂²×C₁₀₉	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

1202 conjugacy classes of subgroups; only the 800 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 800 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 — 5 generators on 125 points:

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

Conjugacy classes & character table

5450 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#8441, C₁₀₉×#12773, C₁₀₉×#17520, …
maximal normal subgroup of	C₁₀₉×#8441, C₁₀₉×#12773, C₁₀₉×#17520, …
maximal quotient of	C₁₀₉×#34129, C₁₀₉×#34155, C₄×#550×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2215);