C₂×#585×C₁₀₉ #28360847

order	13952
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₁₀₉ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#585×C₁₀₉

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶×C₁₀₉
socle	C₂²×C₁₀₉	C₂⁵
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

5956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 6 generators on 127 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#20606, C₁₀₉×#20608, C₁₀₉×#20610, …
maximal normal subgroup of	C₁₀₉×#20606, C₁₀₉×#20608, C₁₀₉×#20610, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2325);

C2×#585×C109 #28360847