C₁₀₉×#2873 #28360127

order	13952
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₁₀₉×#2873

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂⁴
commutator	C₂	C₂²×C₄²×C₁₀₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂³×C₄
prime radical	C₂⁴×C₁₀₉	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 4 generators on 129 points:

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

Conjugacy classes & character table

7412 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#9387, C₁₀₉×#9388, C₁₀₉×#9491, …
maximal normal subgroup of	C₁₀₉×#9387, C₁₀₉×#9388, C₁₀₉×#9491, …
maximal quotient of	C₁₀₉×#17314, C₁₀₉×#17319, C₁₀₉×#17324, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 1605);

C109×#2873 #28360127