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G = SD16×C19order 304 = 24·19

Direct product of C19 and SD16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: SD16×C19, Q8⋊C38, C82C38, D4.C38, C1526C2, C38.15D4, C76.18C22, C4.2(C2×C38), (Q8×C19)⋊4C2, C2.4(D4×C19), (D4×C19).2C2, SmallGroup(304,25)

Series: Derived Chief Lower central Upper central

C1C4 — SD16×C19
C1C2C4C76Q8×C19 — SD16×C19
C1C2C4 — SD16×C19
C1C38C76 — SD16×C19

Generators and relations for SD16×C19
 G = < a,b,c | a19=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

4C2
2C4
2C22
4C38
2C76
2C2×C38

Smallest permutation representation of SD16×C19
On 152 points
Generators in S152
(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)
(1 148 20 57 97 92 131 58)(2 149 21 39 98 93 132 59)(3 150 22 40 99 94 133 60)(4 151 23 41 100 95 115 61)(5 152 24 42 101 77 116 62)(6 134 25 43 102 78 117 63)(7 135 26 44 103 79 118 64)(8 136 27 45 104 80 119 65)(9 137 28 46 105 81 120 66)(10 138 29 47 106 82 121 67)(11 139 30 48 107 83 122 68)(12 140 31 49 108 84 123 69)(13 141 32 50 109 85 124 70)(14 142 33 51 110 86 125 71)(15 143 34 52 111 87 126 72)(16 144 35 53 112 88 127 73)(17 145 36 54 113 89 128 74)(18 146 37 55 114 90 129 75)(19 147 38 56 96 91 130 76)
(20 131)(21 132)(22 133)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 149)(40 150)(41 151)(42 152)(43 134)(44 135)(45 136)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 92)(59 93)(60 94)(61 95)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)

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

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

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

133 conjugacy classes

class 1 2A2B4A4B8A8B19A···19R38A···38R38S···38AJ76A···76R76S···76AJ152A···152AJ
order122448819···1938···3838···3876···7676···76152···152
size11424221···11···14···42···24···42···2

133 irreducible representations

dim111111112222
type+++++
imageC1C2C2C2C19C38C38C38D4SD16D4×C19SD16×C19
kernelSD16×C19C152D4×C19Q8×C19SD16C8D4Q8C38C19C2C1
# reps111118181818121836

Matrix representation of SD16×C19 in GL2(𝔽457) generated by

2560
0256
,
210247
210210
,
10
0456
G:=sub<GL(2,GF(457))| [256,0,0,256],[210,210,247,210],[1,0,0,456] >;

SD16×C19 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_{19}
% in TeX

G:=Group("SD16xC19");
// GroupNames label

G:=SmallGroup(304,25);
// by ID

G=gap.SmallGroup(304,25);
# by ID

G:=PCGroup([5,-2,-2,-19,-2,-2,760,781,4563,2288,58]);
// Polycyclic

G:=Group<a,b,c|a^19=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

Export

Subgroup lattice of SD16×C19 in TeX

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