#72234266

order	4368
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

derived

Sp(2,𝔽₁₃) ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, Fitting, socle, hypercenter	C₂	PGL(2,𝔽₁₃)
commutator, cosocle, hypocenter	Sp(2,𝔽₁₃)	C₂
LCS	G ▷ Sp(2,𝔽₁₃)
UCS	C₁ ◁ C₂

39 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 2 generators on 112 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₇×#72234266
maximal normal subgroup of	C₇×#72234266
maximal quotient of	C₇×#72234266

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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