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## G = Dic13.4D4order 416 = 25·13

### 4th non-split extension by Dic13 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — Dic13.4D4
 Chief series C1 — C13 — C26 — Dic13 — C2×Dic13 — C13⋊M4(2) — Dic13.4D4
 Lower central C13 — C26 — C2×C26 — Dic13.4D4
 Upper central C1 — C2 — C22 — C23

Generators and relations for Dic13.4D4
G = < a,b,c,d | a26=1, b2=c4=a13, d2=b, bab-1=a-1, cac-1=dad-1=a5, cbc-1=a13b, bd=db, dcd-1=a13bc3 >

Smallest permutation representation of Dic13.4D4
On 104 points
Generators in S104
(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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 27 14 40)(2 52 15 39)(3 51 16 38)(4 50 17 37)(5 49 18 36)(6 48 19 35)(7 47 20 34)(8 46 21 33)(9 45 22 32)(10 44 23 31)(11 43 24 30)(12 42 25 29)(13 41 26 28)(53 79 66 92)(54 104 67 91)(55 103 68 90)(56 102 69 89)(57 101 70 88)(58 100 71 87)(59 99 72 86)(60 98 73 85)(61 97 74 84)(62 96 75 83)(63 95 76 82)(64 94 77 81)(65 93 78 80)
(1 91 27 67 14 104 40 54)(2 86 52 72 15 99 39 59)(3 81 51 77 16 94 38 64)(4 102 50 56 17 89 37 69)(5 97 49 61 18 84 36 74)(6 92 48 66 19 79 35 53)(7 87 47 71 20 100 34 58)(8 82 46 76 21 95 33 63)(9 103 45 55 22 90 32 68)(10 98 44 60 23 85 31 73)(11 93 43 65 24 80 30 78)(12 88 42 70 25 101 29 57)(13 83 41 75 26 96 28 62)
(1 104 27 67 14 91 40 54)(2 99 52 72 15 86 39 59)(3 94 51 77 16 81 38 64)(4 89 50 56 17 102 37 69)(5 84 49 61 18 97 36 74)(6 79 48 66 19 92 35 53)(7 100 47 71 20 87 34 58)(8 95 46 76 21 82 33 63)(9 90 45 55 22 103 32 68)(10 85 44 60 23 98 31 73)(11 80 43 65 24 93 30 78)(12 101 42 70 25 88 29 57)(13 96 41 75 26 83 28 62)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 8A 8B 8C 8D 13A 13B 13C 26A ··· 26U order 1 2 2 2 2 4 4 8 8 8 8 13 13 13 26 ··· 26 size 1 1 2 4 52 26 26 52 52 52 52 4 4 4 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 2 4 4 4 4 4 type + + + + + + + + image C1 C2 C2 C4 C4 D4 C4.D4 C13⋊C4 C2×C13⋊C4 D13.D4 Dic13.4D4 kernel Dic13.4D4 C13⋊M4(2) C2×C13⋊D4 C22×D13 C22×C26 Dic13 C13 C23 C22 C2 C1 # reps 1 2 1 2 2 2 1 3 3 6 12

Matrix representation of Dic13.4D4 in GL4(𝔽313) generated by

 64 63 305 261 0 269 0 0 0 0 33 0 0 0 0 19
,
 312 51 0 51 270 1 1 1 0 0 0 1 0 0 312 0
,
 1 0 0 262 0 0 0 312 0 1 0 0 43 312 312 312
,
 312 51 51 51 0 0 0 1 0 1 0 0 270 1 1 1
G:=sub<GL(4,GF(313))| [64,0,0,0,63,269,0,0,305,0,33,0,261,0,0,19],[312,270,0,0,51,1,0,0,0,1,0,312,51,1,1,0],[1,0,0,43,0,0,1,312,0,0,0,312,262,312,0,312],[312,0,0,270,51,0,1,1,51,0,0,1,51,1,0,1] >;

Dic13.4D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{13}._4D_4
% in TeX

G:=Group("Dic13.4D4");
// GroupNames label

G:=SmallGroup(416,88);
// by ID

G=gap.SmallGroup(416,88);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,188,86,579,9221,3473]);
// Polycyclic

G:=Group<a,b,c,d|a^26=1,b^2=c^4=a^13,d^2=b,b*a*b^-1=a^-1,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=a^13*b,b*d=d*b,d*c*d^-1=a^13*b*c^3>;
// generators/relations

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