C₁₀₉×#3407 #28360661

order	13952
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₀₉×#3407

Subgroups

	subgroup	G/H
center	C₈×C₁₀₉	C₂⁴
commutator	C₂	C₂³×C₈×C₁₀₉
Frattini, cosocle	C₈	C₂⁴×C₁₀₉
socle	C₂×C₁₀₉	C₂³×C₈
prime radical	C₂×#41×C₁₀₉	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₉ ◁ G

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 4 generators on 141 points:

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

Conjugacy classes & character table

7412 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#12618, C₁₀₉×#12833, C₁₀₉×#12915, …
maximal normal subgroup of	C₁₀₉×#12618, C₁₀₉×#12833, C₁₀₉×#12915, …
maximal quotient of	C₁₀₉×#33324, C₁₀₉×#33326, C₁₀₉×#33333, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2139);

C109×#3407 #28360661