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## G = C11×D13order 286 = 2·11·13

### Direct product of C11 and D13

Aliases: C11×D13, C13⋊C22, C1432C2, SmallGroup(286,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C11×D13
 Chief series C1 — C13 — C143 — C11×D13
 Lower central C13 — C11×D13
 Upper central C1 — C11

Generators and relations for C11×D13
G = < a,b,c | a11=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C11×D13
On 143 points
Generators in S143
(1 136 126 117 93 85 67 57 43 35 24)(2 137 127 105 94 86 68 58 44 36 25)(3 138 128 106 95 87 69 59 45 37 26)(4 139 129 107 96 88 70 60 46 38 14)(5 140 130 108 97 89 71 61 47 39 15)(6 141 118 109 98 90 72 62 48 27 16)(7 142 119 110 99 91 73 63 49 28 17)(8 143 120 111 100 79 74 64 50 29 18)(9 131 121 112 101 80 75 65 51 30 19)(10 132 122 113 102 81 76 53 52 31 20)(11 133 123 114 103 82 77 54 40 32 21)(12 134 124 115 104 83 78 55 41 33 22)(13 135 125 116 92 84 66 56 42 34 23)
(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)(105 106 107 108 109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140 141 142 143)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 20)(15 19)(16 18)(21 26)(22 25)(23 24)(27 29)(30 39)(31 38)(32 37)(33 36)(34 35)(40 45)(41 44)(42 43)(46 52)(47 51)(48 50)(53 60)(54 59)(55 58)(56 57)(61 65)(62 64)(66 67)(68 78)(69 77)(70 76)(71 75)(72 74)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)(92 93)(94 104)(95 103)(96 102)(97 101)(98 100)(105 115)(106 114)(107 113)(108 112)(109 111)(116 117)(118 120)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 143)

G:=sub<Sym(143)| (1,136,126,117,93,85,67,57,43,35,24)(2,137,127,105,94,86,68,58,44,36,25)(3,138,128,106,95,87,69,59,45,37,26)(4,139,129,107,96,88,70,60,46,38,14)(5,140,130,108,97,89,71,61,47,39,15)(6,141,118,109,98,90,72,62,48,27,16)(7,142,119,110,99,91,73,63,49,28,17)(8,143,120,111,100,79,74,64,50,29,18)(9,131,121,112,101,80,75,65,51,30,19)(10,132,122,113,102,81,76,53,52,31,20)(11,133,123,114,103,82,77,54,40,32,21)(12,134,124,115,104,83,78,55,41,33,22)(13,135,125,116,92,84,66,56,42,34,23), (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)(105,106,107,108,109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,20)(15,19)(16,18)(21,26)(22,25)(23,24)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)(40,45)(41,44)(42,43)(46,52)(47,51)(48,50)(53,60)(54,59)(55,58)(56,57)(61,65)(62,64)(66,67)(68,78)(69,77)(70,76)(71,75)(72,74)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(92,93)(94,104)(95,103)(96,102)(97,101)(98,100)(105,115)(106,114)(107,113)(108,112)(109,111)(116,117)(118,120)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,143)>;

G:=Group( (1,136,126,117,93,85,67,57,43,35,24)(2,137,127,105,94,86,68,58,44,36,25)(3,138,128,106,95,87,69,59,45,37,26)(4,139,129,107,96,88,70,60,46,38,14)(5,140,130,108,97,89,71,61,47,39,15)(6,141,118,109,98,90,72,62,48,27,16)(7,142,119,110,99,91,73,63,49,28,17)(8,143,120,111,100,79,74,64,50,29,18)(9,131,121,112,101,80,75,65,51,30,19)(10,132,122,113,102,81,76,53,52,31,20)(11,133,123,114,103,82,77,54,40,32,21)(12,134,124,115,104,83,78,55,41,33,22)(13,135,125,116,92,84,66,56,42,34,23), (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)(105,106,107,108,109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,20)(15,19)(16,18)(21,26)(22,25)(23,24)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)(40,45)(41,44)(42,43)(46,52)(47,51)(48,50)(53,60)(54,59)(55,58)(56,57)(61,65)(62,64)(66,67)(68,78)(69,77)(70,76)(71,75)(72,74)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(92,93)(94,104)(95,103)(96,102)(97,101)(98,100)(105,115)(106,114)(107,113)(108,112)(109,111)(116,117)(118,120)(121,130)(122,129)(123,128)(124,127)(125,126)(131,140)(132,139)(133,138)(134,137)(135,136)(141,143) );

G=PermutationGroup([(1,136,126,117,93,85,67,57,43,35,24),(2,137,127,105,94,86,68,58,44,36,25),(3,138,128,106,95,87,69,59,45,37,26),(4,139,129,107,96,88,70,60,46,38,14),(5,140,130,108,97,89,71,61,47,39,15),(6,141,118,109,98,90,72,62,48,27,16),(7,142,119,110,99,91,73,63,49,28,17),(8,143,120,111,100,79,74,64,50,29,18),(9,131,121,112,101,80,75,65,51,30,19),(10,132,122,113,102,81,76,53,52,31,20),(11,133,123,114,103,82,77,54,40,32,21),(12,134,124,115,104,83,78,55,41,33,22),(13,135,125,116,92,84,66,56,42,34,23)], [(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),(105,106,107,108,109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140,141,142,143)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,20),(15,19),(16,18),(21,26),(22,25),(23,24),(27,29),(30,39),(31,38),(32,37),(33,36),(34,35),(40,45),(41,44),(42,43),(46,52),(47,51),(48,50),(53,60),(54,59),(55,58),(56,57),(61,65),(62,64),(66,67),(68,78),(69,77),(70,76),(71,75),(72,74),(79,90),(80,89),(81,88),(82,87),(83,86),(84,85),(92,93),(94,104),(95,103),(96,102),(97,101),(98,100),(105,115),(106,114),(107,113),(108,112),(109,111),(116,117),(118,120),(121,130),(122,129),(123,128),(124,127),(125,126),(131,140),(132,139),(133,138),(134,137),(135,136),(141,143)])

88 conjugacy classes

 class 1 2 11A ··· 11J 13A ··· 13F 22A ··· 22J 143A ··· 143BH order 1 2 11 ··· 11 13 ··· 13 22 ··· 22 143 ··· 143 size 1 13 1 ··· 1 2 ··· 2 13 ··· 13 2 ··· 2

88 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C11 C22 D13 C11×D13 kernel C11×D13 C143 D13 C13 C11 C1 # reps 1 1 10 10 6 60

Matrix representation of C11×D13 in GL2(𝔽859) generated by

 61 0 0 61
,
 858 1 610 248
,
 858 0 610 1
G:=sub<GL(2,GF(859))| [61,0,0,61],[858,610,1,248],[858,610,0,1] >;

C11×D13 in GAP, Magma, Sage, TeX

C_{11}\times D_{13}
% in TeX

G:=Group("C11xD13");
// GroupNames label

G:=SmallGroup(286,2);
// by ID

G=gap.SmallGroup(286,2);
# by ID

G:=PCGroup([3,-2,-11,-13,2378]);
// Polycyclic

G:=Group<a,b,c|a^11=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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