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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

C1C2 — D4×C26
C1C2C26C2×C26D4×C13 — D4×C26
C1C2 — D4×C26
C1C2×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 [×2], C2 [×4], C4 [×2], C22, C22 [×4], C22 [×4], C2×C4, D4 [×4], C23 [×2], C13, C2×D4, C26, C26 [×2], C26 [×4], C52 [×2], C2×C26, C2×C26 [×4], C2×C26 [×4], C2×C52, D4×C13 [×4], C22×C26 [×2], D4×C26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C13, C2×D4, C26 [×7], C2×C26 [×7], D4×C13 [×2], 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 40 75 79)(2 41 76 80)(3 42 77 81)(4 43 78 82)(5 44 53 83)(6 45 54 84)(7 46 55 85)(8 47 56 86)(9 48 57 87)(10 49 58 88)(11 50 59 89)(12 51 60 90)(13 52 61 91)(14 27 62 92)(15 28 63 93)(16 29 64 94)(17 30 65 95)(18 31 66 96)(19 32 67 97)(20 33 68 98)(21 34 69 99)(22 35 70 100)(23 36 71 101)(24 37 72 102)(25 38 73 103)(26 39 74 104)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(53 96)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)(61 104)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)

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

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

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

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 2A2B2C2D2E2F2G4A4B13A···13L26A···26AJ26AK···26CF52A···52X
order122222224413···1326···2626···2652···52
size11112222221···11···12···22···2

130 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C13C26C26C26D4D4×C13
kernelD4×C26C2×C52D4×C13C22×C26C2×D4C2×C4D4C23C26C2
# reps114212124824224

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

600
090
009
,
5200
001
0520
,
5200
001
010
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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