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G = C4oD4xC19order 304 = 24·19

Direct product of C19 and C4oD4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C4oD4xC19, D4:2C38, Q8:2C38, C38.13C23, C76.21C22, (C2xC4):3C38, (C2xC76):7C2, (D4xC19):5C2, C4.5(C2xC38), (Q8xC19):5C2, C22.(C2xC38), (C2xC38).2C22, C2.3(C22xC38), SmallGroup(304,40)

Series: Derived Chief Lower central Upper central

C1C2 — C4oD4xC19
C1C2C38C2xC38D4xC19 — C4oD4xC19
C1C2 — C4oD4xC19
C1C76 — C4oD4xC19

Generators and relations for C4oD4xC19
 G = < a,b,c,d | a19=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 46 in 40 conjugacy classes, 34 normal (10 characteristic)
Quotients: C1, C2, C22, C23, C4oD4, C19, C38, C2xC38, C22xC38, C4oD4xC19
2C2
2C2
2C2
2C38
2C38
2C38

Smallest permutation representation of C4oD4xC19
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 36 120 135)(2 37 121 136)(3 38 122 137)(4 20 123 138)(5 21 124 139)(6 22 125 140)(7 23 126 141)(8 24 127 142)(9 25 128 143)(10 26 129 144)(11 27 130 145)(12 28 131 146)(13 29 132 147)(14 30 133 148)(15 31 115 149)(16 32 116 150)(17 33 117 151)(18 34 118 152)(19 35 119 134)(39 64 98 95)(40 65 99 77)(41 66 100 78)(42 67 101 79)(43 68 102 80)(44 69 103 81)(45 70 104 82)(46 71 105 83)(47 72 106 84)(48 73 107 85)(49 74 108 86)(50 75 109 87)(51 76 110 88)(52 58 111 89)(53 59 112 90)(54 60 113 91)(55 61 114 92)(56 62 96 93)(57 63 97 94)
(1 135 120 36)(2 136 121 37)(3 137 122 38)(4 138 123 20)(5 139 124 21)(6 140 125 22)(7 141 126 23)(8 142 127 24)(9 143 128 25)(10 144 129 26)(11 145 130 27)(12 146 131 28)(13 147 132 29)(14 148 133 30)(15 149 115 31)(16 150 116 32)(17 151 117 33)(18 152 118 34)(19 134 119 35)(39 64 98 95)(40 65 99 77)(41 66 100 78)(42 67 101 79)(43 68 102 80)(44 69 103 81)(45 70 104 82)(46 71 105 83)(47 72 106 84)(48 73 107 85)(49 74 108 86)(50 75 109 87)(51 76 110 88)(52 58 111 89)(53 59 112 90)(54 60 113 91)(55 61 114 92)(56 62 96 93)(57 63 97 94)
(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 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(77 136)(78 137)(79 138)(80 139)(81 140)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 134)(95 135)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 115)(113 116)(114 117)

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

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

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

190 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E19A···19R38A···38R38S···38BT76A···76AJ76AK···76CL
order122224444419···1938···3838···3876···7676···76
size11222112221···11···12···21···12···2

190 irreducible representations

dim1111111122
type++++
imageC1C2C2C2C19C38C38C38C4oD4C4oD4xC19
kernelC4oD4xC19C2xC76D4xC19Q8xC19C4oD4C2xC4D4Q8C19C1
# reps133118545418236

Matrix representation of C4oD4xC19 in GL2(F229) generated by

1040
0104
,
1220
0122
,
1070
146122
,
83214
47146
G:=sub<GL(2,GF(229))| [104,0,0,104],[122,0,0,122],[107,146,0,122],[83,47,214,146] >;

C4oD4xC19 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{19}
% in TeX

G:=Group("C4oD4xC19");
// GroupNames label

G:=SmallGroup(304,40);
// by ID

G=gap.SmallGroup(304,40);
# by ID

G:=PCGroup([5,-2,-2,-2,-19,-2,1541,582]);
// Polycyclic

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

Export

Subgroup lattice of C4oD4xC19 in TeX

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