C₁₀₉×C₂²⋊C₃₂ #28358653

order	13952
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₁₀₉×C₂²⋊C₃₂
unambiguous semidirect product	C₂²⋊(C₃₂×C₁₀₉)

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₁₀₉	C₂²
commutator	C₂	C₂×C₃₂×C₁₀₉
Frattini, cosocle	C₂×C₁₆	C₂²×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₁₆
prime radical	C₂³×C₁₀₉	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₀₉ ◁ G

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 145
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 2 generators on 145 points:

(1,2)(3,4)(5,6,7,11,8,12,15,21,9,13,16,22,18,24,27,31,10,14,17,23,19,25,28,32,20,26,29,33,30,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145);(2,4)

Conjugacy classes & character table

8720 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#7334, C₁₀₉×#7335, C₁₀₉×#7362, …
maximal normal subgroup of	C₁₀₉×#7334, C₁₀₉×#7335, C₁₀₉×#7362, …
maximal quotient of	C₁₀₉×#7334, C₁₀₉×#7362, C₁₀₉×D₄⋊C₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 131);

C109×C22⋊C32 #28358653