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G = D26.8D4order 416 = 25·13

4th non-split extension by D26 of D4 acting via D4/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C1042C4, D26.8D4, D13.1SD16, Dic13.2Q8, C82(C13⋊C4), C13⋊(C4.Q8), C132C85C4, C52.8(C2×C4), C26.1(C4⋊C4), (C8×D13).5C2, C52⋊C4.3C2, C2.4(C52⋊C4), (C4×D13).25C22, C4.8(C2×C13⋊C4), SmallGroup(416,68)

Series: Derived Chief Lower central Upper central

C1C52 — D26.8D4
C1C13C26D26C4×D13C52⋊C4 — D26.8D4
C13C26C52 — D26.8D4
C1C2C4C8

Generators and relations for D26.8D4
 G = < a,b,c,d | a26=b2=1, c4=a13, d2=a12b, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c3 >

13C2
13C2
13C4
13C22
52C4
52C4
13C2×C4
13C8
26C2×C4
26C2×C4
4C13⋊C4
4C13⋊C4
13C2×C8
13C4⋊C4
13C4⋊C4
2C2×C13⋊C4
2C2×C13⋊C4
13C4.Q8

Smallest permutation representation of D26.8D4
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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(48 52)(49 51)(53 57)(54 56)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)(100 104)(101 103)
(1 96 44 75 14 83 31 62)(2 97 45 76 15 84 32 63)(3 98 46 77 16 85 33 64)(4 99 47 78 17 86 34 65)(5 100 48 53 18 87 35 66)(6 101 49 54 19 88 36 67)(7 102 50 55 20 89 37 68)(8 103 51 56 21 90 38 69)(9 104 52 57 22 91 39 70)(10 79 27 58 23 92 40 71)(11 80 28 59 24 93 41 72)(12 81 29 60 25 94 42 73)(13 82 30 61 26 95 43 74)
(1 31)(2 52 26 36)(3 47 25 41)(4 42 24 46)(5 37 23 51)(6 32 22 30)(7 27 21 35)(8 48 20 40)(9 43 19 45)(10 38 18 50)(11 33 17 29)(12 28 16 34)(13 49 15 39)(14 44)(53 68 71 56)(54 63 70 61)(55 58 69 66)(57 74 67 76)(59 64 65 60)(62 75)(72 77 78 73)(79 103 87 89)(80 98 86 94)(81 93 85 99)(82 88 84 104)(90 100 102 92)(91 95 101 97)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F8A8B8C8D13A13B13C26A26B26C52A···52F104A···104L
order1222444444888813131326262652···52104···104
size111313226525252522226264444444···44···4

38 irreducible representations

dim111112224444
type+++-+++
imageC1C2C2C4C4Q8D4SD16C13⋊C4C2×C13⋊C4C52⋊C4D26.8D4
kernelD26.8D4C8×D13C52⋊C4C132C8C104Dic13D26D13C8C4C2C1
# reps1122211433612

Matrix representation of D26.8D4 in GL4(𝔽313) generated by

70200199100
2138484213
131230231101
2432132430
,
700284284
2131101101
13110112
243212242241
,
23760311231
217920
80142682
2512510257
,
12653174204
190260160160
140153222192
27028318
G:=sub<GL(4,GF(313))| [70,213,131,243,200,84,230,213,199,84,231,243,100,213,101,0],[70,213,131,243,0,1,101,212,284,101,1,242,284,101,2,241],[237,2,80,251,60,179,142,251,311,2,6,0,231,0,82,257],[126,190,140,27,53,260,153,0,174,160,222,283,204,160,192,18] >;

D26.8D4 in GAP, Magma, Sage, TeX

D_{26}._8D_4
% in TeX

G:=Group("D26.8D4");
// GroupNames label

G:=SmallGroup(416,68);
// by ID

G=gap.SmallGroup(416,68);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,55,579,69,9221,3473]);
// Polycyclic

G:=Group<a,b,c,d|a^26=b^2=1,c^4=a^13,d^2=a^12*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^3>;
// generators/relations

Export

Subgroup lattice of D26.8D4 in TeX

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