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## G = C2×C4×D19order 304 = 24·19

### Direct product of C2×C4 and D19

Aliases: C2×C4×D19, C763C22, C38.2C23, C22.9D38, D38.4C22, Dic193C22, (C2×C76)⋊5C2, C381(C2×C4), C191(C22×C4), (C2×Dic19)⋊5C2, (C2×C38).9C22, C2.1(C22×D19), (C22×D19).2C2, SmallGroup(304,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C2×C4×D19
 Chief series C1 — C19 — C38 — D38 — C22×D19 — C2×C4×D19
 Lower central C19 — C2×C4×D19
 Upper central C1 — C2×C4

Generators and relations for C2×C4×D19
G = < a,b,c,d | a2=b4=c19=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 396 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C22×C4, C19, D19, C38, C38, Dic19, C76, D38, C2×C38, C4×D19, C2×Dic19, C2×C76, C22×D19, C2×C4×D19
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D19, D38, C4×D19, C22×D19, C2×C4×D19

Smallest permutation representation of C2×C4×D19
On 152 points
Generators in S152
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 77)(16 78)(17 79)(18 80)(19 81)(20 114)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 133)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 152)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 140)(66 141)(67 142)(68 143)(69 144)(70 145)(71 146)(72 147)(73 148)(74 149)(75 150)(76 151)
(1 64 20 44)(2 65 21 45)(3 66 22 46)(4 67 23 47)(5 68 24 48)(6 69 25 49)(7 70 26 50)(8 71 27 51)(9 72 28 52)(10 73 29 53)(11 74 30 54)(12 75 31 55)(13 76 32 56)(14 58 33 57)(15 59 34 39)(16 60 35 40)(17 61 36 41)(18 62 37 42)(19 63 38 43)(77 134 109 133)(78 135 110 115)(79 136 111 116)(80 137 112 117)(81 138 113 118)(82 139 114 119)(83 140 96 120)(84 141 97 121)(85 142 98 122)(86 143 99 123)(87 144 100 124)(88 145 101 125)(89 146 102 126)(90 147 103 127)(91 148 104 128)(92 149 105 129)(93 150 106 130)(94 151 107 131)(95 152 108 132)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(39 48)(40 47)(41 46)(42 45)(43 44)(49 57)(50 56)(51 55)(52 54)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(70 76)(71 75)(72 74)(77 86)(78 85)(79 84)(80 83)(81 82)(87 95)(88 94)(89 93)(90 92)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(113 114)(115 122)(116 121)(117 120)(118 119)(123 133)(124 132)(125 131)(126 130)(127 129)(134 143)(135 142)(136 141)(137 140)(138 139)(144 152)(145 151)(146 150)(147 149)

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

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

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

88 conjugacy classes

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

88 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 D19 D38 D38 C4×D19 kernel C2×C4×D19 C4×D19 C2×Dic19 C2×C76 C22×D19 D38 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 8 9 18 9 36

Matrix representation of C2×C4×D19 in GL3(𝔽229) generated by

 228 0 0 0 228 0 0 0 228
,
 1 0 0 0 122 0 0 0 122
,
 1 0 0 0 176 1 0 47 146
,
 228 0 0 0 146 228 0 18 83
G:=sub<GL(3,GF(229))| [228,0,0,0,228,0,0,0,228],[1,0,0,0,122,0,0,0,122],[1,0,0,0,176,47,0,1,146],[228,0,0,0,146,18,0,228,83] >;

C2×C4×D19 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_{19}
% in TeX

G:=Group("C2xC4xD19");
// GroupNames label

G:=SmallGroup(304,28);
// by ID

G=gap.SmallGroup(304,28);
# by ID

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

G:=Group<a,b,c,d|a^2=b^4=c^19=d^2=1,a*b=b*a,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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