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## G = C2×C19⋊C12order 456 = 23·3·19

### Direct product of C2 and C19⋊C12

Aliases: C2×C19⋊C12, C38⋊C12, Dic193C6, (C2×C38).C6, C192(C2×C12), (C2×Dic19)⋊C3, C38.4(C2×C6), C22.(C19⋊C6), (C2×C19⋊C3)⋊C4, C19⋊C32(C2×C4), C2.2(C2×C19⋊C6), (C22×C19⋊C3).C2, (C2×C19⋊C3).4C22, SmallGroup(456,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C2×C19⋊C12
 Chief series C1 — C19 — C38 — C2×C19⋊C3 — C19⋊C12 — C2×C19⋊C12
 Lower central C19 — C2×C19⋊C12
 Upper central C1 — C22

Generators and relations for C2×C19⋊C12
G = < a,b,c | a2=b19=c12=1, ab=ba, ac=ca, cbc-1=b8 >

Smallest permutation representation of C2×C19⋊C12
On 152 points
Generators in S152
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(77 115)(78 116)(79 117)(80 118)(81 119)(82 120)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 143)(97 144)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)(112 140)(113 141)(114 142)
(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 99 20 86)(2 111 31 85 8 107 21 79 12 98 27 94)(3 104 23 84 15 96 22 91 4 97 34 83)(5 109 26 82 10 112 24 77 7 114 29 80)(6 102 37 81 17 101 25 89 18 113 36 88)(9 100 32 78 19 106 28 87 13 110 38 93)(11 105 35 95 14 103 30 92 16 108 33 90)(39 146 58 124)(40 139 69 123 46 135 59 117 50 145 65 132)(41 151 61 122 53 143 60 129 42 144 72 121)(43 137 64 120 48 140 62 115 45 142 67 118)(44 149 75 119 55 148 63 127 56 141 74 126)(47 147 70 116 57 134 66 125 51 138 76 131)(49 152 73 133 52 150 68 130 54 136 71 128)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A ··· 6F 12A ··· 12H 19A 19B 19C 38A ··· 38I order 1 2 2 2 3 3 4 4 4 4 6 ··· 6 12 ··· 12 19 19 19 38 ··· 38 size 1 1 1 1 19 19 19 19 19 19 19 ··· 19 19 ··· 19 6 6 6 6 ··· 6

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 6 6 6 type + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 C19⋊C6 C19⋊C12 C2×C19⋊C6 kernel C2×C19⋊C12 C19⋊C12 C22×C19⋊C3 C2×Dic19 C2×C19⋊C3 Dic19 C2×C38 C38 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 3 6 3

Matrix representation of C2×C19⋊C12 in GL7(𝔽229)

 228 0 0 0 0 0 0 0 228 0 0 0 0 0 0 0 228 0 0 0 0 0 0 0 228 0 0 0 0 0 0 0 228 0 0 0 0 0 0 0 228 0 0 0 0 0 0 0 228
,
 1 0 0 0 0 0 0 0 101 106 83 106 101 228 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 18 0 0 0 0 0 0 0 203 49 159 12 212 139 0 137 15 191 142 128 9 0 165 46 187 198 44 211 0 90 17 217 70 180 81 0 5 58 51 113 36 104 0 176 176 166 175 139 176

G:=sub<GL(7,GF(229))| [228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228,0,0,0,0,0,0,0,228],[1,0,0,0,0,0,0,0,101,1,0,0,0,0,0,106,0,1,0,0,0,0,83,0,0,1,0,0,0,106,0,0,0,1,0,0,101,0,0,0,0,1,0,228,0,0,0,0,0],[18,0,0,0,0,0,0,0,203,137,165,90,5,176,0,49,15,46,17,58,176,0,159,191,187,217,51,166,0,12,142,198,70,113,175,0,212,128,44,180,36,139,0,139,9,211,81,104,176] >;

C2×C19⋊C12 in GAP, Magma, Sage, TeX

C_2\times C_{19}\rtimes C_{12}
% in TeX

G:=Group("C2xC19:C12");
// GroupNames label

G:=SmallGroup(456,10);
// by ID

G=gap.SmallGroup(456,10);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-19,60,10804,1064]);
// Polycyclic

G:=Group<a,b,c|a^2=b^19=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^8>;
// generators/relations

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