C₁₀₉×#1689 #28358943

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#1689

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₁₀₉
prime radical	C₂×D₄×C₁₀₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ G

166 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 133
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 133 points:

(1,2)(3,5)(4,7)(6,8)(9,10,13,16)(11,17,19,24)(12,21,18,15)(14,22,23,20)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(9,11,18,22,13,19,12,20)(10,14,21,24,16,23,15,17);(2,5)(3,8)(6,7)(11,19)(14,23)(17,24)(20,22)

Conjugacy classes & character table

2507 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#12326, C₁₀₉×#12327, C₁₀₉×#12350, …
maximal normal subgroup of	C₁₀₉×#12326, C₁₀₉×#12327, C₁₀₉×#12350, …
maximal quotient of	C₁₀₉×#8652, C₁₀₉×#9227, C₁₀₉×#9569, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 421);