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## G = D4×C26order 208 = 24·13

### Direct product of C26 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C26, C23⋊C26, C524C22, C26.11C23, C4⋊(C2×C26), (C2×C4)⋊2C26, (C2×C52)⋊6C2, C22⋊(C2×C26), (C22×C26)⋊1C2, (C2×C26)⋊2C22, C2.1(C22×C26), SmallGroup(208,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C26
 Chief series C1 — C2 — C26 — C2×C26 — D4×C13 — D4×C26
 Lower central C1 — C2 — D4×C26
 Upper central C1 — C2×C26 — D4×C26

Generators and relations for D4×C26
G = < a,b,c | a26=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C13, C2×D4, C26, C26, C26, C52, C2×C26, C2×C26, C2×C26, C2×C52, D4×C13, C22×C26, D4×C26
Quotients: C1, C2, C22, D4, C23, C13, C2×D4, C26, C2×C26, D4×C13, C22×C26, D4×C26

Smallest permutation representation of D4×C26
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 56 44 93)(2 57 45 94)(3 58 46 95)(4 59 47 96)(5 60 48 97)(6 61 49 98)(7 62 50 99)(8 63 51 100)(9 64 52 101)(10 65 27 102)(11 66 28 103)(12 67 29 104)(13 68 30 79)(14 69 31 80)(15 70 32 81)(16 71 33 82)(17 72 34 83)(18 73 35 84)(19 74 36 85)(20 75 37 86)(21 76 38 87)(22 77 39 88)(23 78 40 89)(24 53 41 90)(25 54 42 91)(26 55 43 92)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 89)(28 90)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(41 103)(42 104)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)

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

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

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

D4×C26 is a maximal subgroup of
D4⋊Dic13  C52.D4  C23⋊Dic13  D526C22  C23.18D26  C52.17D4  C23⋊D26  C522D4  Dic13⋊D4  C52⋊D4  D46D26

130 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 13A ··· 13L 26A ··· 26AJ 26AK ··· 26CF 52A ··· 52X order 1 2 2 2 2 2 2 2 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

130 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C13 C26 C26 C26 D4 D4×C13 kernel D4×C26 C2×C52 D4×C13 C22×C26 C2×D4 C2×C4 D4 C23 C26 C2 # reps 1 1 4 2 12 12 48 24 2 24

Matrix representation of D4×C26 in GL3(𝔽53) generated by

 6 0 0 0 9 0 0 0 9
,
 52 0 0 0 0 1 0 52 0
,
 52 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(53))| [6,0,0,0,9,0,0,0,9],[52,0,0,0,0,52,0,1,0],[52,0,0,0,0,1,0,1,0] >;

D4×C26 in GAP, Magma, Sage, TeX

D_4\times C_{26}
% in TeX

G:=Group("D4xC26");
// GroupNames label

G:=SmallGroup(208,46);
// by ID

G=gap.SmallGroup(208,46);
# by ID

G:=PCGroup([5,-2,-2,-2,-13,-2,1061]);
// Polycyclic

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

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