C₁₀₉×#2205 #28359459

order	13952
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#2205

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂³×C₁₀₉
prime radical	2₋¹⁺⁴×C₁₀₉	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂²×C₁₀₉ ◁ C₂×Q₈×C₁₀₉ ◁ G

224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 1744

A minimal faithful permutation representation — 3 generators on 125 points:

(1,2)(3,11)(4,14)(5,9)(6,16)(7,12)(8,13)(10,15)(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);(1,3,7,6)(2,8,12,14)(4,15,13,5)(9,11,10,16);(1,4,7,13)(2,9,12,10)(3,5,6,15)(8,16,14,11)

Conjugacy classes & character table

2180 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#13683, C₁₀₉×#13684, C₂×C₁₀₉×#2205, …
maximal normal subgroup of	C₁₀₉×#13683, C₁₀₉×#13684, C₂×C₁₀₉×#2205, …
maximal quotient of	C₁₀₉×#12101, C₁₀₉×#12130, C₁₀₉×#12146, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 937);

C109×#2205 #28359459