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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

C1C2 — C13×C4○D4
C1C2C26C2×C26D4×C13 — C13×C4○D4
C1C2 — C13×C4○D4
C1C52 — 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 >

2C2
2C2
2C2
2C26
2C26
2C26

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 2A2B2C2D4A4B4C4D4E13A···13L26A···26L26M···26AV52A···52X52Y···52BH
order122224444413···1326···2626···2652···5252···52
size11222112221···11···12···21···12···2

130 irreducible representations

dim1111111122
type++++
imageC1C2C2C2C13C26C26C26C4○D4C13×C4○D4
kernelC13×C4○D4C2×C52D4×C13Q8×C13C4○D4C2×C4D4Q8C13C1
# reps133112363612224

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

460
046
,
230
023
,
300
2123
,
3723
3516
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

Export

Subgroup lattice of C13×C4○D4 in TeX

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