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G = Dic13.4D4order 416 = 25·13

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic13.4D4, C13⋊(C4.D4), C23.(C13⋊C4), (C22×C26).4C4, C13⋊M4(2)⋊1C2, (C22×D13).2C4, C26.11(C22⋊C4), C2.11(D13.D4), (C2×Dic13).24C22, C22.5(C2×C13⋊C4), (C2×C26).12(C2×C4), (C2×C13⋊D4).8C2, SmallGroup(416,88)

Series: Derived Chief Lower central Upper central

C1C2×C26 — Dic13.4D4
C1C13C26Dic13C2×Dic13C13⋊M4(2) — Dic13.4D4
C13C26C2×C26 — Dic13.4D4
C1C2C22C23

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 >

2C2
4C2
52C2
2C22
4C22
13C4
13C4
26C22
52C22
2C26
4D13
4C26
13C23
13C2×C4
26C8
26D4
26C8
26D4
2D26
2C2×C26
4D26
4C2×C26
13C2×D4
13M4(2)
13M4(2)
2C13⋊D4
2C13⋊C8
2C13⋊C8
2C13⋊D4
13C4.D4

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 2A2B2C2D4A4B8A8B8C8D13A13B13C26A···26U
order1222244888813131326···26
size1124522626525252524444···4

35 irreducible representations

dim11111244444
type++++++++
imageC1C2C2C4C4D4C4.D4C13⋊C4C2×C13⋊C4D13.D4Dic13.4D4
kernelDic13.4D4C13⋊M4(2)C2×C13⋊D4C22×D13C22×C26Dic13C13C23C22C2C1
# reps121222133612

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

6463305261
026900
00330
00019
,
31251051
270111
0001
003120
,
100262
000312
0100
43312312312
,
312515151
0001
0100
270111
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

Export

Subgroup lattice of Dic13.4D4 in TeX

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