C₄×#126×C₁₀₉ #28359558

order	13952
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₄×#126×C₁₀₉

Subgroups

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

516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 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) = 129
transitive degree	μ_t(G) = 6976

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₈×#126×C₁₀₉, C₁₀₉×#21665, C₁₀₉×#21666, …
maximal normal subgroup of	C₈×#126×C₁₀₉, C₁₀₉×#21665, C₁₀₉×#21666, …
maximal quotient of	C₄×#388×C₁₀₉, C₄×#387×C₁₀₉, C₄×#383×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 1036);