C₁₀₉×#2112 #28359366

order	13952
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#2112

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₀₉	C₂³
commutator	C₂²	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂⁴×C₁₀₉	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₀₉ ◁ G

260 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 128 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)(5,6,7,11,8,12,13,18,9,10,14,16,15,17,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(2,4);(6,10)(11,16)(12,17)(18,20)

Conjugacy classes & character table

6104 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#7184, C₁₀₉×#7185, C₁₀₉×#7200, …
maximal normal subgroup of	C₁₀₉×#7184, C₁₀₉×#7185, C₁₀₉×#7200, …
maximal quotient of	C₁₀₉×#11418, C₁₀₉×#11440, C₁₀₉×#11460, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 844);