C₂²×M₃₂×C₁₀₉ #28360659

order	13952
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×M₃₂×C₁₀₉

Subgroups

	subgroup	G/H
center	C₂²×C₈×C₁₀₉	C₂²
commutator	C₂	C₂³×C₈×C₁₀₉
Frattini, cosocle	C₈	C₂⁴×C₁₀₉
socle	C₂³×C₁₀₉	C₂×C₈
prime radical	C₂⁴×C₁₀₉	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₀₉ ◁ G

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 6976

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#12544, C₁₀₉×#12545, C₂×C₁₀₉×#1375, …
maximal normal subgroup of	C₁₀₉×#12544, C₁₀₉×#12545, C₂×C₁₀₉×#1375, …
maximal quotient of	C₂²×#346×C₁₀₉, C₂×C₄×M₃₂×C₁₀₉, C₂²×C₂²⋊C₁₆×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2137);

C22×M32×C109 #28360659