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G = D26.D4order 416 = 25·13

1st non-split extension by D26 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D26.1D4, (C2×C52)⋊1C4, C131(C23⋊C4), (C2×D52).2C2, D13.D41C2, (C22×D13)⋊2C4, C26.1(C22⋊C4), C2.4(D13.D4), (C22×D13).14C22, (C2×C4)⋊(C13⋊C4), (C2×C26).2(C2×C4), C22.2(C2×C13⋊C4), SmallGroup(416,74)

Series: Derived Chief Lower central Upper central

C1C2×C26 — D26.D4
C1C13C26D26C22×D13D13.D4 — D26.D4
C13C26C2×C26 — D26.D4
C1C2C22C2×C4

Generators and relations for D26.D4
 G = < a,b,c,d | a26=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a5, cbc-1=a17b, dbd-1=a4b, dcd-1=a12bc-1 >

2C2
26C2
26C2
52C2
2C4
13C22
13C22
26C22
52C4
52C4
52C22
52C22
2D13
2D13
2C26
4D13
13C23
13C23
26C2×C4
26D4
26D4
26C2×C4
2C52
2D26
4D26
4C13⋊C4
4C13⋊C4
4D26
13C2×D4
13C22⋊C4
13C22⋊C4
2C2×C13⋊C4
2C2×C13⋊C4
2D52
2D52
13C23⋊C4

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

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E13A13B13C26A···26I52A···52L
order1222224444413131326···2652···52
size1122626524525252524444···44···4

35 irreducible representations

dim11111244444
type+++++++++
imageC1C2C2C4C4D4C23⋊C4C13⋊C4C2×C13⋊C4D13.D4D26.D4
kernelD26.D4D13.D4C2×D52C2×C52C22×D13D26C13C2×C4C22C2C1
# reps121222133612

Matrix representation of D26.D4 in GL4(𝔽53) generated by

12391146
7887
46113912
4115498
,
12391146
8491541
4653845
17417
,
1629475
23184328
47171428
4117195
,
1000
748158
1238445
0010
G:=sub<GL(4,GF(53))| [12,7,46,41,39,8,11,15,11,8,39,49,46,7,12,8],[12,8,46,1,39,49,5,7,11,15,38,41,46,41,45,7],[16,23,47,41,29,18,17,17,47,43,14,19,5,28,28,5],[1,7,12,0,0,48,38,0,0,15,4,1,0,8,45,0] >;

D26.D4 in GAP, Magma, Sage, TeX

D_{26}.D_4
% in TeX

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

G:=SmallGroup(416,74);
// by ID

G=gap.SmallGroup(416,74);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,188,579,9221,3473]);
// Polycyclic

G:=Group<a,b,c,d|a^26=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^5,c*b*c^-1=a^17*b,d*b*d^-1=a^4*b,d*c*d^-1=a^12*b*c^-1>;
// generators/relations

Export

Subgroup lattice of D26.D4 in TeX

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