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## G = C10×C13⋊C3order 390 = 2·3·5·13

### Direct product of C10 and C13⋊C3

Aliases: C10×C13⋊C3, C130⋊C3, C26⋊C15, C654C6, C132C30, SmallGroup(390,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C10×C13⋊C3
 Chief series C1 — C13 — C65 — C5×C13⋊C3 — C10×C13⋊C3
 Lower central C13 — C10×C13⋊C3
 Upper central C1 — C10

Generators and relations for C10×C13⋊C3
G = < a,b,c | a10=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >

Smallest permutation representation of C10×C13⋊C3
On 130 points
Generators in S130
(1 92 53 79 40 66 27 118 14 105)(2 93 54 80 41 67 28 119 15 106)(3 94 55 81 42 68 29 120 16 107)(4 95 56 82 43 69 30 121 17 108)(5 96 57 83 44 70 31 122 18 109)(6 97 58 84 45 71 32 123 19 110)(7 98 59 85 46 72 33 124 20 111)(8 99 60 86 47 73 34 125 21 112)(9 100 61 87 48 74 35 126 22 113)(10 101 62 88 49 75 36 127 23 114)(11 102 63 89 50 76 37 128 24 115)(12 103 64 90 51 77 38 129 25 116)(13 104 65 91 52 78 39 130 26 117)
(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)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)(54 56 62)(55 59 58)(57 65 63)(60 61 64)(67 69 75)(68 72 71)(70 78 76)(73 74 77)(80 82 88)(81 85 84)(83 91 89)(86 87 90)(93 95 101)(94 98 97)(96 104 102)(99 100 103)(106 108 114)(107 111 110)(109 117 115)(112 113 116)(119 121 127)(120 124 123)(122 130 128)(125 126 129)

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

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

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

70 conjugacy classes

 class 1 2 3A 3B 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 13A 13B 13C 13D 15A ··· 15H 26A 26B 26C 26D 30A ··· 30H 65A ··· 65P 130A ··· 130P order 1 2 3 3 5 5 5 5 6 6 10 10 10 10 13 13 13 13 15 ··· 15 26 26 26 26 30 ··· 30 65 ··· 65 130 ··· 130 size 1 1 13 13 1 1 1 1 13 13 1 1 1 1 3 3 3 3 13 ··· 13 3 3 3 3 13 ··· 13 3 ··· 3 3 ··· 3

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C5 C6 C10 C15 C30 C13⋊C3 C2×C13⋊C3 C5×C13⋊C3 C10×C13⋊C3 kernel C10×C13⋊C3 C5×C13⋊C3 C130 C2×C13⋊C3 C65 C13⋊C3 C26 C13 C10 C5 C2 C1 # reps 1 1 2 4 2 4 8 8 4 4 16 16

Matrix representation of C10×C13⋊C3 in GL3(𝔽1171) generated by

 955 0 0 0 955 0 0 0 955
,
 760 21 1 1 0 0 0 1 0
,
 1 0 0 1149 759 21 852 22 411
G:=sub<GL(3,GF(1171))| [955,0,0,0,955,0,0,0,955],[760,1,0,21,0,1,1,0,0],[1,1149,852,0,759,22,0,21,411] >;

C10×C13⋊C3 in GAP, Magma, Sage, TeX

C_{10}\times C_{13}\rtimes C_3
% in TeX

G:=Group("C10xC13:C3");
// GroupNames label

G:=SmallGroup(390,4);
// by ID

G=gap.SmallGroup(390,4);
# by ID

G:=PCGroup([4,-2,-3,-5,-13,727]);
// Polycyclic

G:=Group<a,b,c|a^10=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

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