#72210865

order	2640
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

derived

SL(2,𝔽₁₁) ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, Fitting, socle, hypercenter	C₂	PGL(2,𝔽₁₁)
commutator, cosocle, hypocenter	SL(2,𝔽₁₁)	C₂
LCS	G ▷ SL(2,𝔽₁₁)
UCS	C₁ ◁ C₂

42 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#72210865
maximal normal subgroup of	C₅×#72210865
maximal quotient of	C₅×#72210865

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

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