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## G = Q8×D19order 304 = 24·19

### Direct product of Q8 and D19

Aliases: Q8×D19, C4.6D38, Dic384C2, C38.7C23, C76.6C22, D38.5C22, Dic19.3C22, C192(C2×Q8), (Q8×C19)⋊2C2, (C4×D19).1C2, C2.8(C22×D19), SmallGroup(304,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C38 — Q8×D19
 Chief series C1 — C19 — C38 — D38 — C4×D19 — Q8×D19
 Lower central C19 — C38 — Q8×D19
 Upper central C1 — C2 — Q8

Generators and relations for Q8×D19
G = < a,b,c,d | a4=c19=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of Q8×D19
On 152 points
Generators in S152
(1 60 37 50)(2 61 38 51)(3 62 20 52)(4 63 21 53)(5 64 22 54)(6 65 23 55)(7 66 24 56)(8 67 25 57)(9 68 26 39)(10 69 27 40)(11 70 28 41)(12 71 29 42)(13 72 30 43)(14 73 31 44)(15 74 32 45)(16 75 33 46)(17 76 34 47)(18 58 35 48)(19 59 36 49)(77 116 107 136)(78 117 108 137)(79 118 109 138)(80 119 110 139)(81 120 111 140)(82 121 112 141)(83 122 113 142)(84 123 114 143)(85 124 96 144)(86 125 97 145)(87 126 98 146)(88 127 99 147)(89 128 100 148)(90 129 101 149)(91 130 102 150)(92 131 103 151)(93 132 104 152)(94 133 105 134)(95 115 106 135)
(1 108 37 78)(2 109 38 79)(3 110 20 80)(4 111 21 81)(5 112 22 82)(6 113 23 83)(7 114 24 84)(8 96 25 85)(9 97 26 86)(10 98 27 87)(11 99 28 88)(12 100 29 89)(13 101 30 90)(14 102 31 91)(15 103 32 92)(16 104 33 93)(17 105 34 94)(18 106 35 95)(19 107 36 77)(39 145 68 125)(40 146 69 126)(41 147 70 127)(42 148 71 128)(43 149 72 129)(44 150 73 130)(45 151 74 131)(46 152 75 132)(47 134 76 133)(48 135 58 115)(49 136 59 116)(50 137 60 117)(51 138 61 118)(52 139 62 119)(53 140 63 120)(54 141 64 121)(55 142 65 122)(56 143 66 123)(57 144 67 124)
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 38)(19 37)(39 70)(40 69)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 60)(50 59)(51 58)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)(85 100)(86 99)(87 98)(88 97)(89 96)(90 114)(91 113)(92 112)(93 111)(94 110)(95 109)(115 138)(116 137)(117 136)(118 135)(119 134)(120 152)(121 151)(122 150)(123 149)(124 148)(125 147)(126 146)(127 145)(128 144)(129 143)(130 142)(131 141)(132 140)(133 139)

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

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

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

55 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 19A ··· 19I 38A ··· 38I 76A ··· 76AA order 1 2 2 2 4 4 4 4 4 4 19 ··· 19 38 ··· 38 76 ··· 76 size 1 1 19 19 2 2 2 38 38 38 2 ··· 2 2 ··· 2 4 ··· 4

55 irreducible representations

 dim 1 1 1 1 2 2 2 4 type + + + + - + + - image C1 C2 C2 C2 Q8 D19 D38 Q8×D19 kernel Q8×D19 Dic38 C4×D19 Q8×C19 D19 Q8 C4 C1 # reps 1 3 3 1 2 9 27 9

Matrix representation of Q8×D19 in GL4(𝔽229) generated by

 1 0 0 0 0 1 0 0 0 0 1 221 0 0 172 228
,
 1 0 0 0 0 1 0 0 0 0 27 142 0 0 140 202
,
 185 1 0 0 82 97 0 0 0 0 1 0 0 0 0 1
,
 97 228 0 0 19 132 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(229))| [1,0,0,0,0,1,0,0,0,0,1,172,0,0,221,228],[1,0,0,0,0,1,0,0,0,0,27,140,0,0,142,202],[185,82,0,0,1,97,0,0,0,0,1,0,0,0,0,1],[97,19,0,0,228,132,0,0,0,0,1,0,0,0,0,1] >;

Q8×D19 in GAP, Magma, Sage, TeX

Q_8\times D_{19}
% in TeX

G:=Group("Q8xD19");
// GroupNames label

G:=SmallGroup(304,33);
// by ID

G=gap.SmallGroup(304,33);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-19,46,97,42,7204]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^19=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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