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## G = D5×SL2(𝔽3)  order 240 = 24·3·5

### Direct product of D5 and SL2(𝔽3)

Aliases: D5×SL2(𝔽3), D10.A4, (Q8×D5)⋊C3, (C5×Q8)⋊C6, Q8⋊(C3×D5), C2.3(D5×A4), C10.2(C2×A4), C5⋊(C2×SL2(𝔽3)), (C5×SL2(𝔽3))⋊3C2, SmallGroup(240,109)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C5×Q8 — D5×SL2(𝔽3)
 Chief series C1 — C2 — C10 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3)
 Lower central C5×Q8 — D5×SL2(𝔽3)
 Upper central C1 — C2

Generators and relations for D5×SL2(𝔽3)
G = < a,b,c,d,e | a5=b2=c4=e3=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Character table of D5×SL2(𝔽3)

 class 1 2A 2B 2C 3A 3B 4A 4B 5A 5B 6A 6B 6C 6D 6E 6F 10A 10B 15A 15B 15C 15D 20A 20B 30A 30B 30C 30D size 1 1 5 5 4 4 6 30 2 2 4 4 20 20 20 20 2 2 8 8 8 8 12 12 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ4 1 1 -1 -1 ζ3 ζ32 1 -1 1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 1 1 ζ32 ζ3 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 linear of order 6 ρ5 1 1 -1 -1 ζ32 ζ3 1 -1 1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 1 1 ζ3 ζ32 ζ32 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ7 2 2 0 0 2 2 2 0 -1-√5/2 -1+√5/2 2 2 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ8 2 2 0 0 2 2 2 0 -1+√5/2 -1-√5/2 2 2 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 -2 -2 2 -1 -1 0 0 2 2 1 1 1 1 -1 -1 -2 -2 -1 -1 -1 -1 0 0 1 1 1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ10 2 -2 2 -2 -1 -1 0 0 2 2 1 1 -1 -1 1 1 -2 -2 -1 -1 -1 -1 0 0 1 1 1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ11 2 -2 -2 2 ζ65 ζ6 0 0 2 2 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 -2 -2 ζ6 ζ65 ζ65 ζ6 0 0 ζ32 ζ3 ζ32 ζ3 complex lifted from SL2(𝔽3) ρ12 2 -2 -2 2 ζ6 ζ65 0 0 2 2 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 -2 -2 ζ65 ζ6 ζ6 ζ65 0 0 ζ3 ζ32 ζ3 ζ32 complex lifted from SL2(𝔽3) ρ13 2 -2 2 -2 ζ65 ζ6 0 0 2 2 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 -2 -2 ζ6 ζ65 ζ65 ζ6 0 0 ζ32 ζ3 ζ32 ζ3 complex lifted from SL2(𝔽3) ρ14 2 -2 2 -2 ζ6 ζ65 0 0 2 2 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 -2 -2 ζ65 ζ6 ζ6 ζ65 0 0 ζ3 ζ32 ζ3 ζ32 complex lifted from SL2(𝔽3) ρ15 2 2 0 0 -1-√-3 -1+√-3 2 0 -1+√5/2 -1-√5/2 -1+√-3 -1-√-3 0 0 0 0 -1-√5/2 -1+√5/2 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 -1+√5/2 -1-√5/2 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 complex lifted from C3×D5 ρ16 2 2 0 0 -1+√-3 -1-√-3 2 0 -1-√5/2 -1+√5/2 -1-√-3 -1+√-3 0 0 0 0 -1+√5/2 -1-√5/2 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 -1-√5/2 -1+√5/2 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 complex lifted from C3×D5 ρ17 2 2 0 0 -1-√-3 -1+√-3 2 0 -1-√5/2 -1+√5/2 -1+√-3 -1-√-3 0 0 0 0 -1+√5/2 -1-√5/2 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 -1-√5/2 -1+√5/2 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 complex lifted from C3×D5 ρ18 2 2 0 0 -1+√-3 -1-√-3 2 0 -1+√5/2 -1-√5/2 -1-√-3 -1+√-3 0 0 0 0 -1-√5/2 -1+√5/2 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 -1+√5/2 -1-√5/2 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 complex lifted from C3×D5 ρ19 3 3 3 3 0 0 -1 -1 3 3 0 0 0 0 0 0 3 3 0 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ20 3 3 -3 -3 0 0 -1 1 3 3 0 0 0 0 0 0 3 3 0 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from C2×A4 ρ21 4 -4 0 0 -2 -2 0 0 -1-√5 -1+√5 2 2 0 0 0 0 1-√5 1+√5 1+√5/2 1-√5/2 1+√5/2 1-√5/2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 symplectic faithful, Schur index 2 ρ22 4 -4 0 0 -2 -2 0 0 -1+√5 -1-√5 2 2 0 0 0 0 1+√5 1-√5 1-√5/2 1+√5/2 1-√5/2 1+√5/2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 symplectic faithful, Schur index 2 ρ23 4 -4 0 0 1-√-3 1+√-3 0 0 -1-√5 -1+√5 -1-√-3 -1+√-3 0 0 0 0 1-√5 1+√5 -ζ32ζ53-ζ32ζ52 -ζ3ζ54-ζ3ζ5 -ζ3ζ53-ζ3ζ52 -ζ32ζ54-ζ32ζ5 0 0 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 complex faithful ρ24 4 -4 0 0 1-√-3 1+√-3 0 0 -1+√5 -1-√5 -1-√-3 -1+√-3 0 0 0 0 1+√5 1-√5 -ζ32ζ54-ζ32ζ5 -ζ3ζ53-ζ3ζ52 -ζ3ζ54-ζ3ζ5 -ζ32ζ53-ζ32ζ52 0 0 ζ32ζ53+ζ32ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ3ζ53+ζ3ζ52 complex faithful ρ25 4 -4 0 0 1+√-3 1-√-3 0 0 -1+√5 -1-√5 -1+√-3 -1-√-3 0 0 0 0 1+√5 1-√5 -ζ3ζ54-ζ3ζ5 -ζ32ζ53-ζ32ζ52 -ζ32ζ54-ζ32ζ5 -ζ3ζ53-ζ3ζ52 0 0 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 complex faithful ρ26 4 -4 0 0 1+√-3 1-√-3 0 0 -1-√5 -1+√5 -1+√-3 -1-√-3 0 0 0 0 1-√5 1+√5 -ζ3ζ53-ζ3ζ52 -ζ32ζ54-ζ32ζ5 -ζ32ζ53-ζ32ζ52 -ζ3ζ54-ζ3ζ5 0 0 ζ3ζ54+ζ3ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ32ζ54+ζ32ζ5 complex faithful ρ27 6 6 0 0 0 0 -2 0 -3-3√5/2 -3+3√5/2 0 0 0 0 0 0 -3+3√5/2 -3-3√5/2 0 0 0 0 1+√5/2 1-√5/2 0 0 0 0 orthogonal lifted from D5×A4 ρ28 6 6 0 0 0 0 -2 0 -3+3√5/2 -3-3√5/2 0 0 0 0 0 0 -3-3√5/2 -3+3√5/2 0 0 0 0 1-√5/2 1+√5/2 0 0 0 0 orthogonal lifted from D5×A4

Smallest permutation representation of D5×SL2(𝔽3)
On 40 points
Generators in S40
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)
(1 12)(2 11)(3 15)(4 14)(5 13)(6 38)(7 37)(8 36)(9 40)(10 39)(16 23)(17 22)(18 21)(19 25)(20 24)(26 33)(27 32)(28 31)(29 35)(30 34)
(1 23 13 17)(2 24 14 18)(3 25 15 19)(4 21 11 20)(5 22 12 16)(6 30 40 31)(7 26 36 32)(8 27 37 33)(9 28 38 34)(10 29 39 35)
(1 8 13 37)(2 9 14 38)(3 10 15 39)(4 6 11 40)(5 7 12 36)(16 26 22 32)(17 27 23 33)(18 28 24 34)(19 29 25 35)(20 30 21 31)
(6 21 31)(7 22 32)(8 23 33)(9 24 34)(10 25 35)(16 26 36)(17 27 37)(18 28 38)(19 29 39)(20 30 40)

G:=sub<Sym(40)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40), (1,12)(2,11)(3,15)(4,14)(5,13)(6,38)(7,37)(8,36)(9,40)(10,39)(16,23)(17,22)(18,21)(19,25)(20,24)(26,33)(27,32)(28,31)(29,35)(30,34), (1,23,13,17)(2,24,14,18)(3,25,15,19)(4,21,11,20)(5,22,12,16)(6,30,40,31)(7,26,36,32)(8,27,37,33)(9,28,38,34)(10,29,39,35), (1,8,13,37)(2,9,14,38)(3,10,15,39)(4,6,11,40)(5,7,12,36)(16,26,22,32)(17,27,23,33)(18,28,24,34)(19,29,25,35)(20,30,21,31), (6,21,31)(7,22,32)(8,23,33)(9,24,34)(10,25,35)(16,26,36)(17,27,37)(18,28,38)(19,29,39)(20,30,40)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40), (1,12)(2,11)(3,15)(4,14)(5,13)(6,38)(7,37)(8,36)(9,40)(10,39)(16,23)(17,22)(18,21)(19,25)(20,24)(26,33)(27,32)(28,31)(29,35)(30,34), (1,23,13,17)(2,24,14,18)(3,25,15,19)(4,21,11,20)(5,22,12,16)(6,30,40,31)(7,26,36,32)(8,27,37,33)(9,28,38,34)(10,29,39,35), (1,8,13,37)(2,9,14,38)(3,10,15,39)(4,6,11,40)(5,7,12,36)(16,26,22,32)(17,27,23,33)(18,28,24,34)(19,29,25,35)(20,30,21,31), (6,21,31)(7,22,32)(8,23,33)(9,24,34)(10,25,35)(16,26,36)(17,27,37)(18,28,38)(19,29,39)(20,30,40) );

G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,38),(7,37),(8,36),(9,40),(10,39),(16,23),(17,22),(18,21),(19,25),(20,24),(26,33),(27,32),(28,31),(29,35),(30,34)], [(1,23,13,17),(2,24,14,18),(3,25,15,19),(4,21,11,20),(5,22,12,16),(6,30,40,31),(7,26,36,32),(8,27,37,33),(9,28,38,34),(10,29,39,35)], [(1,8,13,37),(2,9,14,38),(3,10,15,39),(4,6,11,40),(5,7,12,36),(16,26,22,32),(17,27,23,33),(18,28,24,34),(19,29,25,35),(20,30,21,31)], [(6,21,31),(7,22,32),(8,23,33),(9,24,34),(10,25,35),(16,26,36),(17,27,37),(18,28,38),(19,29,39),(20,30,40)]])

D5×SL2(𝔽3) is a maximal subgroup of   D10.S4  D10.1S4  D10.2S4  SL2(𝔽3).11D10  D20.A4

Matrix representation of D5×SL2(𝔽3) in GL4(𝔽61) generated by

 0 1 0 0 60 17 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 60 0
,
 1 0 0 0 0 1 0 0 0 0 47 13 0 0 13 14
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 47 13
G:=sub<GL(4,GF(61))| [0,60,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,47,13,0,0,13,14],[1,0,0,0,0,1,0,0,0,0,1,47,0,0,0,13] >;

D5×SL2(𝔽3) in GAP, Magma, Sage, TeX

D_5\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("D5xSL(2,3)");
// GroupNames label

G:=SmallGroup(240,109);
// by ID

G=gap.SmallGroup(240,109);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-5,-2,170,374,81,543,261,2884]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=e^3=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations

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