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G = C13×C4≀C2order 416 = 25·13

Direct product of C13 and C4≀C2

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C13×C4≀C2, D42C52, Q82C52, C423C26, C52.66D4, M4(2)⋊4C26, (C4×C52)⋊10C2, (D4×C13)⋊8C4, C4.3(C2×C52), (Q8×C13)⋊8C4, C52.51(C2×C4), C4○D4.1C26, C4.17(D4×C13), (C2×C26).22D4, C22.3(D4×C13), C26.37(C22⋊C4), (C13×M4(2))⋊10C2, (C2×C52).116C22, (C2×C4).19(C2×C26), (C13×C4○D4).4C2, C2.8(C13×C22⋊C4), SmallGroup(416,54)

Series: Derived Chief Lower central Upper central

C1C4 — C13×C4≀C2
C1C2C4C2×C4C2×C52C13×M4(2) — C13×C4≀C2
C1C2C4 — C13×C4≀C2
C1C52C2×C52 — C13×C4≀C2

Generators and relations for C13×C4≀C2
 G = < a,b,c,d | a13=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

2C2
4C2
2C4
2C22
2C4
2C4
2C26
4C26
2D4
2C2×C4
2C2×C4
2C8
2C2×C26
2C52
2C52
2C52
2C2×C52
2C104
2D4×C13
2C2×C52

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

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

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

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

182 conjugacy classes

class 1 2A2B2C4A4B4C···4G4H8A8B13A···13L26A···26L26M···26X26Y···26AJ52A···52X52Y···52CF52CG···52CR104A···104X
order1222444···448813···1326···2626···2626···2652···5252···5252···52104···104
size1124112···24441···11···12···24···41···12···24···44···4

182 irreducible representations

dim111111111111222222
type++++++
imageC1C2C2C2C4C4C13C26C26C26C52C52D4D4C4≀C2D4×C13D4×C13C13×C4≀C2
kernelC13×C4≀C2C4×C52C13×M4(2)C13×C4○D4D4×C13Q8×C13C4≀C2C42M4(2)C4○D4D4Q8C52C2×C26C13C4C22C1
# reps111122121212122424114121248

Matrix representation of C13×C4≀C2 in GL2(𝔽313) generated by

2770
0277
,
2880
025
,
025
2880
,
250
01
G:=sub<GL(2,GF(313))| [277,0,0,277],[288,0,0,25],[0,288,25,0],[25,0,0,1] >;

C13×C4≀C2 in GAP, Magma, Sage, TeX

C_{13}\times C_4\wr C_2
% in TeX

G:=Group("C13xC4wrC2");
// GroupNames label

G:=SmallGroup(416,54);
// by ID

G=gap.SmallGroup(416,54);
# by ID

G:=PCGroup([6,-2,-2,-13,-2,-2,-2,624,649,6243,3129,117,88]);
// Polycyclic

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

Export

Subgroup lattice of C13×C4≀C2 in TeX

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