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G = C3×Dic39order 468 = 22·32·13

Direct product of C3 and Dic39

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3×Dic39, C399C12, C78.9C6, C78.4S3, C6.4D39, C394Dic3, C322Dic13, (C3×C39)⋊8C4, C6.(C3×D13), C2.(C3×D39), C3⋊(C3×Dic13), C26.3(C3×S3), (C3×C78).2C2, (C3×C6).1D13, C135(C3×Dic3), SmallGroup(468,25)

Series: Derived Chief Lower central Upper central

C1C39 — C3×Dic39
C1C13C39C78C3×C78 — C3×Dic39
C39 — C3×Dic39
C1C6

Generators and relations for C3×Dic39
 G = < a,b,c | a3=b78=1, c2=b39, ab=ba, ac=ca, cbc-1=b-1 >

2C3
39C4
2C6
2C39
13Dic3
39C12
3Dic13
2C78
13C3×Dic3
3C3×Dic13

Smallest permutation representation of C3×Dic39
On 156 points
Generators in S156
(1 27 53)(2 28 54)(3 29 55)(4 30 56)(5 31 57)(6 32 58)(7 33 59)(8 34 60)(9 35 61)(10 36 62)(11 37 63)(12 38 64)(13 39 65)(14 40 66)(15 41 67)(16 42 68)(17 43 69)(18 44 70)(19 45 71)(20 46 72)(21 47 73)(22 48 74)(23 49 75)(24 50 76)(25 51 77)(26 52 78)(79 131 105)(80 132 106)(81 133 107)(82 134 108)(83 135 109)(84 136 110)(85 137 111)(86 138 112)(87 139 113)(88 140 114)(89 141 115)(90 142 116)(91 143 117)(92 144 118)(93 145 119)(94 146 120)(95 147 121)(96 148 122)(97 149 123)(98 150 124)(99 151 125)(100 152 126)(101 153 127)(102 154 128)(103 155 129)(104 156 130)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)
(1 92 40 131)(2 91 41 130)(3 90 42 129)(4 89 43 128)(5 88 44 127)(6 87 45 126)(7 86 46 125)(8 85 47 124)(9 84 48 123)(10 83 49 122)(11 82 50 121)(12 81 51 120)(13 80 52 119)(14 79 53 118)(15 156 54 117)(16 155 55 116)(17 154 56 115)(18 153 57 114)(19 152 58 113)(20 151 59 112)(21 150 60 111)(22 149 61 110)(23 148 62 109)(24 147 63 108)(25 146 64 107)(26 145 65 106)(27 144 66 105)(28 143 67 104)(29 142 68 103)(30 141 69 102)(31 140 70 101)(32 139 71 100)(33 138 72 99)(34 137 73 98)(35 136 74 97)(36 135 75 96)(37 134 76 95)(38 133 77 94)(39 132 78 93)

G:=sub<Sym(156)| (1,27,53)(2,28,54)(3,29,55)(4,30,56)(5,31,57)(6,32,58)(7,33,59)(8,34,60)(9,35,61)(10,36,62)(11,37,63)(12,38,64)(13,39,65)(14,40,66)(15,41,67)(16,42,68)(17,43,69)(18,44,70)(19,45,71)(20,46,72)(21,47,73)(22,48,74)(23,49,75)(24,50,76)(25,51,77)(26,52,78)(79,131,105)(80,132,106)(81,133,107)(82,134,108)(83,135,109)(84,136,110)(85,137,111)(86,138,112)(87,139,113)(88,140,114)(89,141,115)(90,142,116)(91,143,117)(92,144,118)(93,145,119)(94,146,120)(95,147,121)(96,148,122)(97,149,123)(98,150,124)(99,151,125)(100,152,126)(101,153,127)(102,154,128)(103,155,129)(104,156,130), (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)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,92,40,131)(2,91,41,130)(3,90,42,129)(4,89,43,128)(5,88,44,127)(6,87,45,126)(7,86,46,125)(8,85,47,124)(9,84,48,123)(10,83,49,122)(11,82,50,121)(12,81,51,120)(13,80,52,119)(14,79,53,118)(15,156,54,117)(16,155,55,116)(17,154,56,115)(18,153,57,114)(19,152,58,113)(20,151,59,112)(21,150,60,111)(22,149,61,110)(23,148,62,109)(24,147,63,108)(25,146,64,107)(26,145,65,106)(27,144,66,105)(28,143,67,104)(29,142,68,103)(30,141,69,102)(31,140,70,101)(32,139,71,100)(33,138,72,99)(34,137,73,98)(35,136,74,97)(36,135,75,96)(37,134,76,95)(38,133,77,94)(39,132,78,93)>;

G:=Group( (1,27,53)(2,28,54)(3,29,55)(4,30,56)(5,31,57)(6,32,58)(7,33,59)(8,34,60)(9,35,61)(10,36,62)(11,37,63)(12,38,64)(13,39,65)(14,40,66)(15,41,67)(16,42,68)(17,43,69)(18,44,70)(19,45,71)(20,46,72)(21,47,73)(22,48,74)(23,49,75)(24,50,76)(25,51,77)(26,52,78)(79,131,105)(80,132,106)(81,133,107)(82,134,108)(83,135,109)(84,136,110)(85,137,111)(86,138,112)(87,139,113)(88,140,114)(89,141,115)(90,142,116)(91,143,117)(92,144,118)(93,145,119)(94,146,120)(95,147,121)(96,148,122)(97,149,123)(98,150,124)(99,151,125)(100,152,126)(101,153,127)(102,154,128)(103,155,129)(104,156,130), (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)(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,92,40,131)(2,91,41,130)(3,90,42,129)(4,89,43,128)(5,88,44,127)(6,87,45,126)(7,86,46,125)(8,85,47,124)(9,84,48,123)(10,83,49,122)(11,82,50,121)(12,81,51,120)(13,80,52,119)(14,79,53,118)(15,156,54,117)(16,155,55,116)(17,154,56,115)(18,153,57,114)(19,152,58,113)(20,151,59,112)(21,150,60,111)(22,149,61,110)(23,148,62,109)(24,147,63,108)(25,146,64,107)(26,145,65,106)(27,144,66,105)(28,143,67,104)(29,142,68,103)(30,141,69,102)(31,140,70,101)(32,139,71,100)(33,138,72,99)(34,137,73,98)(35,136,74,97)(36,135,75,96)(37,134,76,95)(38,133,77,94)(39,132,78,93) );

G=PermutationGroup([(1,27,53),(2,28,54),(3,29,55),(4,30,56),(5,31,57),(6,32,58),(7,33,59),(8,34,60),(9,35,61),(10,36,62),(11,37,63),(12,38,64),(13,39,65),(14,40,66),(15,41,67),(16,42,68),(17,43,69),(18,44,70),(19,45,71),(20,46,72),(21,47,73),(22,48,74),(23,49,75),(24,50,76),(25,51,77),(26,52,78),(79,131,105),(80,132,106),(81,133,107),(82,134,108),(83,135,109),(84,136,110),(85,137,111),(86,138,112),(87,139,113),(88,140,114),(89,141,115),(90,142,116),(91,143,117),(92,144,118),(93,145,119),(94,146,120),(95,147,121),(96,148,122),(97,149,123),(98,150,124),(99,151,125),(100,152,126),(101,153,127),(102,154,128),(103,155,129),(104,156,130)], [(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),(79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)], [(1,92,40,131),(2,91,41,130),(3,90,42,129),(4,89,43,128),(5,88,44,127),(6,87,45,126),(7,86,46,125),(8,85,47,124),(9,84,48,123),(10,83,49,122),(11,82,50,121),(12,81,51,120),(13,80,52,119),(14,79,53,118),(15,156,54,117),(16,155,55,116),(17,154,56,115),(18,153,57,114),(19,152,58,113),(20,151,59,112),(21,150,60,111),(22,149,61,110),(23,148,62,109),(24,147,63,108),(25,146,64,107),(26,145,65,106),(27,144,66,105),(28,143,67,104),(29,142,68,103),(30,141,69,102),(31,140,70,101),(32,139,71,100),(33,138,72,99),(34,137,73,98),(35,136,74,97),(36,135,75,96),(37,134,76,95),(38,133,77,94),(39,132,78,93)])

126 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E12A12B12C12D13A···13F26A···26F39A···39AV78A···78AV
order123333344666661212121213···1326···2639···3978···78
size1111222393911222393939392···22···22···22···2

126 irreducible representations

dim111111222222222222
type+++-+-+-
imageC1C2C3C4C6C12S3Dic3C3×S3D13C3×Dic3Dic13C3×D13D39C3×Dic13Dic39C3×D39C3×Dic39
kernelC3×Dic39C3×C78Dic39C3×C39C78C39C78C39C26C3×C6C13C32C6C6C3C3C2C1
# reps112224112626121212122424

Matrix representation of C3×Dic39 in GL3(𝔽157) generated by

1200
0120
0012
,
15600
0710
00115
,
12900
001
010
G:=sub<GL(3,GF(157))| [12,0,0,0,12,0,0,0,12],[156,0,0,0,71,0,0,0,115],[129,0,0,0,0,1,0,1,0] >;

C3×Dic39 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{39}
% in TeX

G:=Group("C3xDic39");
// GroupNames label

G:=SmallGroup(468,25);
// by ID

G=gap.SmallGroup(468,25);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-13,30,483,10804]);
// Polycyclic

G:=Group<a,b,c|a^3=b^78=1,c^2=b^39,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic39 in TeX

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