C₁₀₉×#1536 #28358790

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₉×#1536

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₄²	C₂³×C₁₀₉
prime radical	#532×C₁₀₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

238 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 133
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 133 points:

(1,2,3,6,4,5,7,8)(9,10)(11,18,20,23)(12,21)(13,16)(14,17)(15,19,24,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(2,5)(3,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)

Conjugacy classes & character table

2834 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#7484, C₁₀₉×#7494, C₁₀₉×#8475, …
maximal normal subgroup of	C₁₀₉×#7484, C₁₀₉×#7494, C₁₀₉×#8475, …
maximal quotient of	C₁₀₉×#7597, C₁₀₉×#7603, C₁₀₉×#7605, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 268);