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## G = Dic5.D4order 160 = 25·5

### 1st non-split extension by Dic5 of D4 acting via D4/C2=C22

Aliases: Dic5.1D4, (C2×C4).F5, (C2×C20).1C4, C5⋊(C4.10D4), C22.F5.C2, C22.3(C2×F5), (C2×Dic5).2C4, C2.5(C22⋊F5), C10.3(C22⋊C4), (C2×Dic10).2C2, (C2×Dic5).23C22, (C2×C10).8(C2×C4), SmallGroup(160,80)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic5.D4
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C22.F5 — Dic5.D4
 Lower central C5 — C10 — C2×C10 — Dic5.D4
 Upper central C1 — C2 — C22 — C2×C4

Generators and relations for Dic5.D4
G = < a,b,c,d | a10=1, b2=c4=a5, d2=b, bab-1=a-1, cac-1=dad-1=a3, cbc-1=a5b, bd=db, dcd-1=bc3 >

Character table of Dic5.D4

 class 1 2A 2B 4A 4B 4C 4D 5 8A 8B 8C 8D 10A 10B 10C 20A 20B 20C 20D size 1 1 2 4 10 10 20 4 20 20 20 20 4 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 1 1 1 -1 1 1 -1 1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 -1 -1 -1 1 -i -i i i 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 -1 -1 -1 1 1 i -i -i i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 i i -i -i 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 -1 -1 -1 1 1 -i i i -i 1 1 1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 -2 0 -2 2 0 2 0 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 0 2 -2 0 2 0 0 0 0 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 0 0 0 0 -1 0 0 0 0 -1 1 1 √5 -√5 -√5 √5 orthogonal lifted from C22⋊F5 ρ12 4 4 4 -4 0 0 0 -1 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ13 4 4 4 4 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 4 -4 0 0 0 0 -1 0 0 0 0 -1 1 1 -√5 √5 √5 -√5 orthogonal lifted from C22⋊F5 ρ15 4 -4 0 0 0 0 0 4 0 0 0 0 -4 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ16 4 -4 0 0 0 0 0 -1 0 0 0 0 1 -√5 √5 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ4ζ52+2ζ4ζ5+ζ4 symplectic faithful, Schur index 2 ρ17 4 -4 0 0 0 0 0 -1 0 0 0 0 1 √5 -√5 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ43ζ54+2ζ43ζ52+ζ43 symplectic faithful, Schur index 2 ρ18 4 -4 0 0 0 0 0 -1 0 0 0 0 1 √5 -√5 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ4ζ54+2ζ4ζ53+ζ4 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ43ζ53+2ζ43ζ5+ζ43 symplectic faithful, Schur index 2 ρ19 4 -4 0 0 0 0 0 -1 0 0 0 0 1 -√5 √5 2ζ4ζ52+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ52+ζ43 2ζ43ζ53+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ53+ζ4 symplectic faithful, Schur index 2

Smallest permutation representation of Dic5.D4
On 80 points
Generators in S80
(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)(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)
(1 16 6 11)(2 15 7 20)(3 14 8 19)(4 13 9 18)(5 12 10 17)(21 39 26 34)(22 38 27 33)(23 37 28 32)(24 36 29 31)(25 35 30 40)(41 54 46 59)(42 53 47 58)(43 52 48 57)(44 51 49 56)(45 60 50 55)(61 79 66 74)(62 78 67 73)(63 77 68 72)(64 76 69 71)(65 75 70 80)
(1 73 11 62 6 78 16 67)(2 80 20 65 7 75 15 70)(3 77 19 68 8 72 14 63)(4 74 18 61 9 79 13 66)(5 71 17 64 10 76 12 69)(21 57 34 43 26 52 39 48)(22 54 33 46 27 59 38 41)(23 51 32 49 28 56 37 44)(24 58 31 42 29 53 36 47)(25 55 40 45 30 60 35 50)
(1 58 16 42 6 53 11 47)(2 55 15 45 7 60 20 50)(3 52 14 48 8 57 19 43)(4 59 13 41 9 54 18 46)(5 56 12 44 10 51 17 49)(21 77 39 68 26 72 34 63)(22 74 38 61 27 79 33 66)(23 71 37 64 28 76 32 69)(24 78 36 67 29 73 31 62)(25 75 35 70 30 80 40 65)

G:=sub<Sym(80)| (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)(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), (1,16,6,11)(2,15,7,20)(3,14,8,19)(4,13,9,18)(5,12,10,17)(21,39,26,34)(22,38,27,33)(23,37,28,32)(24,36,29,31)(25,35,30,40)(41,54,46,59)(42,53,47,58)(43,52,48,57)(44,51,49,56)(45,60,50,55)(61,79,66,74)(62,78,67,73)(63,77,68,72)(64,76,69,71)(65,75,70,80), (1,73,11,62,6,78,16,67)(2,80,20,65,7,75,15,70)(3,77,19,68,8,72,14,63)(4,74,18,61,9,79,13,66)(5,71,17,64,10,76,12,69)(21,57,34,43,26,52,39,48)(22,54,33,46,27,59,38,41)(23,51,32,49,28,56,37,44)(24,58,31,42,29,53,36,47)(25,55,40,45,30,60,35,50), (1,58,16,42,6,53,11,47)(2,55,15,45,7,60,20,50)(3,52,14,48,8,57,19,43)(4,59,13,41,9,54,18,46)(5,56,12,44,10,51,17,49)(21,77,39,68,26,72,34,63)(22,74,38,61,27,79,33,66)(23,71,37,64,28,76,32,69)(24,78,36,67,29,73,31,62)(25,75,35,70,30,80,40,65)>;

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)(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), (1,16,6,11)(2,15,7,20)(3,14,8,19)(4,13,9,18)(5,12,10,17)(21,39,26,34)(22,38,27,33)(23,37,28,32)(24,36,29,31)(25,35,30,40)(41,54,46,59)(42,53,47,58)(43,52,48,57)(44,51,49,56)(45,60,50,55)(61,79,66,74)(62,78,67,73)(63,77,68,72)(64,76,69,71)(65,75,70,80), (1,73,11,62,6,78,16,67)(2,80,20,65,7,75,15,70)(3,77,19,68,8,72,14,63)(4,74,18,61,9,79,13,66)(5,71,17,64,10,76,12,69)(21,57,34,43,26,52,39,48)(22,54,33,46,27,59,38,41)(23,51,32,49,28,56,37,44)(24,58,31,42,29,53,36,47)(25,55,40,45,30,60,35,50), (1,58,16,42,6,53,11,47)(2,55,15,45,7,60,20,50)(3,52,14,48,8,57,19,43)(4,59,13,41,9,54,18,46)(5,56,12,44,10,51,17,49)(21,77,39,68,26,72,34,63)(22,74,38,61,27,79,33,66)(23,71,37,64,28,76,32,69)(24,78,36,67,29,73,31,62)(25,75,35,70,30,80,40,65) );

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),(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)], [(1,16,6,11),(2,15,7,20),(3,14,8,19),(4,13,9,18),(5,12,10,17),(21,39,26,34),(22,38,27,33),(23,37,28,32),(24,36,29,31),(25,35,30,40),(41,54,46,59),(42,53,47,58),(43,52,48,57),(44,51,49,56),(45,60,50,55),(61,79,66,74),(62,78,67,73),(63,77,68,72),(64,76,69,71),(65,75,70,80)], [(1,73,11,62,6,78,16,67),(2,80,20,65,7,75,15,70),(3,77,19,68,8,72,14,63),(4,74,18,61,9,79,13,66),(5,71,17,64,10,76,12,69),(21,57,34,43,26,52,39,48),(22,54,33,46,27,59,38,41),(23,51,32,49,28,56,37,44),(24,58,31,42,29,53,36,47),(25,55,40,45,30,60,35,50)], [(1,58,16,42,6,53,11,47),(2,55,15,45,7,60,20,50),(3,52,14,48,8,57,19,43),(4,59,13,41,9,54,18,46),(5,56,12,44,10,51,17,49),(21,77,39,68,26,72,34,63),(22,74,38,61,27,79,33,66),(23,71,37,64,28,76,32,69),(24,78,36,67,29,73,31,62),(25,75,35,70,30,80,40,65)]])

Dic5.D4 is a maximal subgroup of
C42.F5  C42.2F5  (D4×C10).C4  (Q8×C10).C4  (C4×D5).D4  (C2×D4).9F5  (C2×Q8).7F5  Dic5.4D12  (C2×C60).C4
Dic5.D4 is a maximal quotient of
C10.C4≀C2  Dic5.D8  (C2×C20)⋊1C8  (C22×C4).F5  C22.F5⋊C4  Dic5.4D12  (C2×C60).C4

Matrix representation of Dic5.D4 in GL8(𝔽41)

 34 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 21 0 0 1 0 0 0 0 21 0 40 6 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 15 21 0 0 0 0 0 0 3 26 0 0 0 0 0 0 2 28 25 16 0 0 0 0 0 26 2 16 0 0 0 0 0 0 0 0 0 37 0 0 0 0 0 0 31 0 0 0 0 0 0 0 4 24 1 21 0 0 0 0 21 25 37 40
,
 31 35 26 1 0 0 0 0 27 13 32 14 0 0 0 0 2 19 37 2 0 0 0 0 25 37 23 1 0 0 0 0 0 0 0 0 10 37 7 22 0 0 0 0 30 26 31 21 0 0 0 0 33 38 24 11 0 0 0 0 37 17 6 22
,
 10 6 15 40 0 0 0 0 14 28 9 27 0 0 0 0 39 22 4 39 0 0 0 0 16 4 18 40 0 0 0 0 0 0 0 0 2 0 37 0 0 0 0 0 4 26 1 21 0 0 0 0 1 1 39 0 0 0 0 0 0 40 25 15

G:=sub<GL(8,GF(41))| [34,40,21,21,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[15,3,2,0,0,0,0,0,21,26,28,26,0,0,0,0,0,0,25,2,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,0,0,31,4,21,0,0,0,0,37,0,24,25,0,0,0,0,0,0,1,37,0,0,0,0,0,0,21,40],[31,27,2,25,0,0,0,0,35,13,19,37,0,0,0,0,26,32,37,23,0,0,0,0,1,14,2,1,0,0,0,0,0,0,0,0,10,30,33,37,0,0,0,0,37,26,38,17,0,0,0,0,7,31,24,6,0,0,0,0,22,21,11,22],[10,14,39,16,0,0,0,0,6,28,22,4,0,0,0,0,15,9,4,18,0,0,0,0,40,27,39,40,0,0,0,0,0,0,0,0,2,4,1,0,0,0,0,0,0,26,1,40,0,0,0,0,37,1,39,25,0,0,0,0,0,21,0,15] >;

Dic5.D4 in GAP, Magma, Sage, TeX

{\rm Dic}_5.D_4
% in TeX

G:=Group("Dic5.D4");
// GroupNames label

G:=SmallGroup(160,80);
// by ID

G=gap.SmallGroup(160,80);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,188,86,579,2309,1169]);
// Polycyclic

G:=Group<a,b,c,d|a^10=1,b^2=c^4=a^5,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=b*c^3>;
// generators/relations

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