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## G = C13×C4○D4order 208 = 24·13

### Direct product of C13 and C4○D4

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

Aliases: C13×C4○D4, D42C26, Q82C26, C52.21C22, C26.13C23, (C2×C52)⋊7C2, (C2×C4)⋊3C26, C52(D4×C13), C52(Q8×C13), (D4×C13)⋊5C2, C4.5(C2×C26), (Q8×C13)⋊5C2, C22.(C2×C26), (C2×C26).2C22, C2.3(C22×C26), SmallGroup(208,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C13×C4○D4
 Chief series C1 — C2 — C26 — C2×C26 — D4×C13 — C13×C4○D4
 Lower central C1 — C2 — C13×C4○D4
 Upper central C1 — C52 — C13×C4○D4

Generators and relations for C13×C4○D4
G = < a,b,c,d | a13=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

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

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

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

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

C13×C4○D4 is a maximal subgroup of   C52.56D4  D4.Dic13  D4⋊D26  C52.C23  D4.9D26  D48D26  D4.10D26
C13×C4○D4 is a maximal quotient of   D4×C52  Q8×C52

130 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 13A ··· 13L 26A ··· 26L 26M ··· 26AV 52A ··· 52X 52Y ··· 52BH order 1 2 2 2 2 4 4 4 4 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 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 C4○D4 C13×C4○D4 kernel C13×C4○D4 C2×C52 D4×C13 Q8×C13 C4○D4 C2×C4 D4 Q8 C13 C1 # reps 1 3 3 1 12 36 36 12 2 24

Matrix representation of C13×C4○D4 in GL2(𝔽53) generated by

 46 0 0 46
,
 23 0 0 23
,
 30 0 21 23
,
 37 23 35 16
G:=sub<GL(2,GF(53))| [46,0,0,46],[23,0,0,23],[30,21,0,23],[37,35,23,16] >;

C13×C4○D4 in GAP, Magma, Sage, TeX

C_{13}\times C_4\circ D_4
% in TeX

G:=Group("C13xC4oD4");
// GroupNames label

G:=SmallGroup(208,48);
// by ID

G=gap.SmallGroup(208,48);
# by ID

G:=PCGroup([5,-2,-2,-2,-13,-2,1061,402]);
// Polycyclic

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

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