C₁₀₉×#1338 #28358592

order	13952
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₀₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₁₀₉×#1338

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₈×C₁₀₉
Frattini, cosocle	C₄×C₈	C₂²×C₁₀₉
socle, prime radical	C₂²×C₁₀₉	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₄×C₈×C₁₀₉ ◁ G

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 13952

A minimal faithful permutation representation — 2 generators on 133 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₀₉×#1338, C₁₀₉×#11610, C₁₀₉×#11611, …
maximal normal subgroup of	C₂×C₁₀₉×#1338, C₁₀₉×#11610, C₁₀₉×#11611, …
maximal quotient of	C₁₀₉×#7118, C₁₀₉×#7128, C₁₀₉×#7170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 70);

C109×#1338 #28358592