C₂²×Q₃₂×C₁₀₉ #28360664

order	13952
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×Q₃₂×C₁₀₉

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₁₀₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ G

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 145
transitive degree	μ_t(G) = 13952

A minimal faithful permutation representation — 4 generators on 145 points:

(1,2)(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);(3,4);(5,6,10,14)(7,15,18,34)(8,19,23,33)(9,24,21,13)(11,26,28,16)(12,29,31,22)(17,35,25,30)(20,32,36,27);(5,7,10,18)(6,11,14,28)(8,20,23,36)(9,25,21,17)(12,30,31,35)(13,32,24,27)(15,33,34,19)(16,29,26,22)

Conjugacy classes & character table

4796 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#13015, C₁₀₉×#13020, C₁₀₉×#13236, …
maximal normal subgroup of	C₁₀₉×#13015, C₁₀₉×#13020, C₁₀₉×#13236, …
maximal quotient of	C₂²×#358×C₁₀₉, C₂²×#366×C₁₀₉, C₂×C₄×Q₃₂×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2142);

C22×Q32×C109 #28360664