C₄×#104×C₁₀₉ #28359012

order	13952
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₄×#104×C₁₀₉

Subgroups

	subgroup	G/H
center	C₄²×C₁₀₉	D₄
commutator	C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂×#41×C₁₀₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

360 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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);(5,6,8,11,9,7,12,10);(5,7)(6,9)(8,10)(11,12)

Conjugacy classes & character table

6104 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₈×#104×C₁₀₉, C₁₀₉×#8191, C₁₀₉×#8192, …
maximal normal subgroup of	C₈×#104×C₁₀₉, C₁₀₉×#8191, C₁₀₉×#8192, …
maximal quotient of	C₄×D₄⋊C₈×C₁₀₉, C₄×Q₈⋊C₈×C₁₀₉, C₁₀₉×#8172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 490);