C₁₀₉×#1300 #28358554

order	13952
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₀₉×#1300

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂²⋊C₄
commutator	C₂×C₄	C₄²×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂²×C₁₀₉
socle	C₂²×C₁₀₉	#95
prime radical	C₂³×C₁₀₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

172 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 125
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 2 generators on 125 points:

(1,2)(3,5)(4,7)(6,8)(9,10,13,15,14,11,12,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(2,5,7,6)(9,11,14,10)(12,15,13,16)

Conjugacy classes & character table

4142 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₀₉×#1300, C₁₀₉×#8068, C₁₀₉×#8069, …
maximal normal subgroup of	C₂×C₁₀₉×#1300, C₁₀₉×#8068, C₁₀₉×#8069, …
maximal quotient of	C₁₀₉×#7027, C₁₀₉×#7028, C₁₀₉×#7030, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 32);