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G = C13×M4(2)  order 208 = 24·13

Direct product of C13 and M4(2)

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

Aliases: C13×M4(2), C4.C52, C83C26, C1047C2, C52.7C4, C22.C52, C52.22C22, (C2×C4).2C26, C2.3(C2×C52), (C2×C52).8C2, C4.6(C2×C26), (C2×C26).3C4, C26.19(C2×C4), SmallGroup(208,24)

Series: Derived Chief Lower central Upper central

C1C2 — C13×M4(2)
C1C2C4C52C104 — C13×M4(2)
C1C2 — C13×M4(2)
C1C52 — C13×M4(2)

Generators and relations for C13×M4(2)
 G = < a,b,c | a13=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

2C2
2C26

Smallest permutation representation of C13×M4(2)
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 20 103 54 38 74 40 87)(2 21 104 55 39 75 41 88)(3 22 92 56 27 76 42 89)(4 23 93 57 28 77 43 90)(5 24 94 58 29 78 44 91)(6 25 95 59 30 66 45 79)(7 26 96 60 31 67 46 80)(8 14 97 61 32 68 47 81)(9 15 98 62 33 69 48 82)(10 16 99 63 34 70 49 83)(11 17 100 64 35 71 50 84)(12 18 101 65 36 72 51 85)(13 19 102 53 37 73 52 86)
(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 66)(26 67)(53 86)(54 87)(55 88)(56 89)(57 90)(58 91)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)

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

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

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

C13×M4(2) is a maximal subgroup of   C52.53D4  C52.46D4  C4.12D52  D527C4  D52.2C4  C8⋊D26  C8.D26

130 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D13A···13L26A···26L26M···26X52A···52X52Y···52AJ104A···104AV
order122444888813···1326···2626···2652···5252···52104···104
size11211222221···11···12···21···12···22···2

130 irreducible representations

dim111111111122
type+++
imageC1C2C2C4C4C13C26C26C52C52M4(2)C13×M4(2)
kernelC13×M4(2)C104C2×C52C52C2×C26M4(2)C8C2×C4C4C22C13C1
# reps121221224122424224

Matrix representation of C13×M4(2) in GL2(𝔽313) generated by

2800
0280
,
253311
6660
,
10
253312
G:=sub<GL(2,GF(313))| [280,0,0,280],[253,66,311,60],[1,253,0,312] >;

C13×M4(2) in GAP, Magma, Sage, TeX

C_{13}\times M_4(2)
% in TeX

G:=Group("C13xM4(2)");
// GroupNames label

G:=SmallGroup(208,24);
// by ID

G=gap.SmallGroup(208,24);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-2,260,1061,58]);
// Polycyclic

G:=Group<a,b,c|a^13=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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Subgroup lattice of C13×M4(2) in TeX

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