C₁₀₉×#1482 #28358736

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

Aliases

direct product

C₁₀₉×#1482

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₄²	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	#121×C₁₀₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

212 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 141
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 141 points:

(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,25)(22,28)(26,31)(29,32)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#8489, C₁₀₉×#8490, C₁₀₉×#8514, …
maximal normal subgroup of	C₁₀₉×#8489, C₁₀₉×#8490, C₁₀₉×#8514, …
maximal quotient of	C₁₀₉×#7656, C₁₀₉×#7657, C₁₀₉×#7658, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 214);