C₁₀₉×#1518 #28358772

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₉×#1518

Subgroups

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

220 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 3488

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

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

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#8430, C₁₀₉×#8431, C₁₀₉×#8432, …
maximal normal subgroup of	C₁₀₉×#8430, C₁₀₉×#8431, C₁₀₉×#8432, …
maximal quotient of	C₁₀₉×#7577, C₁₀₉×#7590, C₁₀₉×#7591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 250);