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G = C8⋊D26order 416 = 25·13

1st semidirect product of C8 and D26 acting via D26/C13=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C81D26, D1042C2, C4.14D52, C52.12D4, C1041C22, D524C22, C22.5D52, M4(2)⋊1D13, C52.32C23, Dic264C22, (C2×D52)⋊7C2, (C2×C26).5D4, C104⋊C21C2, C131(C8⋊C22), C26.13(C2×D4), C2.15(C2×D52), (C2×C4).15D26, D525C22C2, (C13×M4(2))⋊1C2, (C2×C52).27C22, C4.30(C22×D13), SmallGroup(416,129)

Series: Derived Chief Lower central Upper central

C1C52 — C8⋊D26
C1C13C26C52D52C2×D52 — C8⋊D26
C13C26C52 — C8⋊D26
C1C2C2×C4M4(2)

Generators and relations for C8⋊D26
 G = < a,b,c | a8=b26=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >

Subgroups: 680 in 68 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2 [×4], C4 [×2], C4, C22, C22 [×5], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C13, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, D13 [×3], C26, C26, C8⋊C22, Dic13, C52 [×2], D26 [×5], C2×C26, C104 [×2], Dic26, C4×D13, D52, D52 [×2], D52, C13⋊D4, C2×C52, C22×D13, C104⋊C2 [×2], D104 [×2], C13×M4(2), C2×D52, D525C2, C8⋊D26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C8⋊C22, D26 [×3], D52 [×2], C22×D13, C2×D52, C8⋊D26

Smallest permutation representation of C8⋊D26
On 104 points
Generators in S104
(1 91 40 61 15 104 37 74)(2 79 41 75 16 92 38 62)(3 93 42 63 17 80 39 76)(4 81 43 77 18 94 27 64)(5 95 44 65 19 82 28 78)(6 83 45 53 20 96 29 66)(7 97 46 67 21 84 30 54)(8 85 47 55 22 98 31 68)(9 99 48 69 23 86 32 56)(10 87 49 57 24 100 33 70)(11 101 50 71 25 88 34 58)(12 89 51 59 26 102 35 72)(13 103 52 73 14 90 36 60)
(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 3)(4 13)(5 12)(6 11)(7 10)(8 9)(14 18)(15 17)(19 26)(20 25)(21 24)(22 23)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(53 88)(54 87)(55 86)(56 85)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)

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

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

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

71 conjugacy classes

class 1 2A2B2C2D2E4A4B4C8A8B13A···13F26A···26F26G···26L52A···52L52M···52R104A···104X
order1222224448813···1326···2626···2652···5252···52104···104
size1125252522252442···22···24···42···24···44···4

71 irreducible representations

dim111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2D4D4D13D26D26D52D52C8⋊C22C8⋊D26
kernelC8⋊D26C104⋊C2D104C13×M4(2)C2×D52D525C2C52C2×C26M4(2)C8C2×C4C4C22C13C1
# reps1221111161261212112

Matrix representation of C8⋊D26 in GL4(𝔽313) generated by

2383032185
117143166304
27827611070
1631558135
,
21430000
7012300
88772913
273248291260
,
127100
14818600
1297289227
61625424
G:=sub<GL(4,GF(313))| [238,117,278,163,303,143,276,15,218,166,110,58,5,304,70,135],[214,70,88,273,300,123,77,248,0,0,29,291,0,0,13,260],[127,148,129,61,1,186,7,62,0,0,289,54,0,0,227,24] >;

C8⋊D26 in GAP, Magma, Sage, TeX

C_8\rtimes D_{26}
% in TeX

G:=Group("C8:D26");
// GroupNames label

G:=SmallGroup(416,129);
// by ID

G=gap.SmallGroup(416,129);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,188,50,579,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^8=b^26=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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