D₄×#41×C₁₀₉ #28360722

order	13952
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₁₀₉ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

D₄×#41×C₁₀₉

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
socle	C₂²×C₁₀₉	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

1648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 5 generators on 121 points:

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

Conjugacy classes & character table

5450 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#8436, C₁₀₉×#17507, C₁₀₉×#17511, …
maximal normal subgroup of	C₁₀₉×#8436, C₁₀₉×#17507, C₁₀₉×#17511, …
maximal quotient of	C₂²⋊C₄×#41×C₁₀₉, C₄⋊C₄×#41×C₁₀₉, D₄×#117×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2200);

D4×#41×C109 #28360722