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G = C8×D19order 304 = 24·19

Direct product of C8 and D19

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C8×D19, C1523C2, D38.2C4, C4.12D38, C76.12C22, Dic19.2C4, C19⋊C86C2, C191(C2×C8), C38.1(C2×C4), C2.1(C4×D19), (C4×D19).3C2, SmallGroup(304,3)

Series: Derived Chief Lower central Upper central

C1C19 — C8×D19
C1C19C38C76C4×D19 — C8×D19
C19 — C8×D19
C1C8

Generators and relations for C8×D19
 G = < a,b,c | a8=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

19C2
19C2
19C22
19C4
19C2×C4
19C8
19C2×C8

Smallest permutation representation of C8×D19
On 152 points
Generators in S152
(1 144 76 102 21 127 40 88)(2 145 58 103 22 128 41 89)(3 146 59 104 23 129 42 90)(4 147 60 105 24 130 43 91)(5 148 61 106 25 131 44 92)(6 149 62 107 26 132 45 93)(7 150 63 108 27 133 46 94)(8 151 64 109 28 115 47 95)(9 152 65 110 29 116 48 77)(10 134 66 111 30 117 49 78)(11 135 67 112 31 118 50 79)(12 136 68 113 32 119 51 80)(13 137 69 114 33 120 52 81)(14 138 70 96 34 121 53 82)(15 139 71 97 35 122 54 83)(16 140 72 98 36 123 55 84)(17 141 73 99 37 124 56 85)(18 142 74 100 38 125 57 86)(19 143 75 101 20 126 39 87)
(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)
(1 20)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(39 76)(40 75)(41 74)(42 73)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(77 112)(78 111)(79 110)(80 109)(81 108)(82 107)(83 106)(84 105)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(91 98)(92 97)(93 96)(94 114)(95 113)(115 136)(116 135)(117 134)(118 152)(119 151)(120 150)(121 149)(122 148)(123 147)(124 146)(125 145)(126 144)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)

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

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

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

88 conjugacy classes

class 1 2A2B2C4A4B4C4D8A8B8C8D8E8F8G8H19A···19I38A···38I76A···76R152A···152AJ
order122244448888888819···1938···3876···76152···152
size1119191119191111191919192···22···22···22···2

88 irreducible representations

dim11111112222
type++++++
imageC1C2C2C2C4C4C8D19D38C4×D19C8×D19
kernelC8×D19C19⋊C8C152C4×D19Dic19D38D19C8C4C2C1
# reps1111228991836

Matrix representation of C8×D19 in GL2(𝔽457) generated by

1700
0170
,
1001
394301
,
1561
343301
G:=sub<GL(2,GF(457))| [170,0,0,170],[100,394,1,301],[156,343,1,301] >;

C8×D19 in GAP, Magma, Sage, TeX

C_8\times D_{19}
% in TeX

G:=Group("C8xD19");
// GroupNames label

G:=SmallGroup(304,3);
// by ID

G=gap.SmallGroup(304,3);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-19,26,42,7204]);
// Polycyclic

G:=Group<a,b,c|a^8=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8×D19 in TeX

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