C₁₀₉×#2231 #28359485

order	13952
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₀₉×#2231

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#476
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₀₉
prime radical	#135×C₁₀₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ C₄×C₈×C₁₀₉ ◁ G

120 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 141
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 3 generators on 141 points:

(1,2,7,12)(3,13,16,11)(4,17)(5,21,14,10)(6,8,23,25)(9,26)(15,29)(18,31)(19,30)(20,24)(22,27)(28,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,3,14,23,7,16,5,6)(2,8,21,11,12,25,10,13)(4,15,19,32,20,26,18,22)(9,24,28,30,29,17,27,31);(1,4)(2,9)(3,15)(5,18)(6,22)(7,20)(8,24)(10,27)(11,30)(12,29)(13,31)(14,19)(16,26)(17,25)(21,28)(23,32)

Conjugacy classes & character table

2507 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#12463, C₁₀₉×#12464, C₁₀₉×#12465, …
maximal normal subgroup of	C₁₀₉×#12463, C₁₀₉×#12464, C₁₀₉×#12465, …
maximal quotient of	C₁₀₉×#12824, C₁₀₉×#12950, C₁₀₉×#13116, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 963);