C₁₀₉×#1708 #28358962

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₁₀₉×#1708

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₁₀₉
prime radical	#127×C₁₀₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ G

190 conjugacy classes of subgroups.

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:

(2,5)(3,7)(6,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,23)(22,24)(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);(9,11,12,18,13,19,21,24)(10,14,16,17,15,22,20,23);(1,2,3,5,4,6,7,8)(10,15)(14,22)(16,20)(17,23)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#7536, C₁₀₉×#12365, C₂×C₁₀₉×#1708, …
maximal normal subgroup of	C₁₀₉×#7536, C₁₀₉×#12365, C₂×C₁₀₉×#1708, …
maximal quotient of	C₁₀₉×#8815, C₁₀₉×#9819, C₁₀₉×#9820, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 440);

C109×#1708 #28358962