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## G = C4○D4×C19order 304 = 24·19

### Direct product of C19 and C4○D4

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

Aliases: C4○D4×C19, D42C38, Q82C38, C38.13C23, C76.21C22, (C2×C4)⋊3C38, (C2×C76)⋊7C2, (D4×C19)⋊5C2, C4.5(C2×C38), (Q8×C19)⋊5C2, C22.(C2×C38), (C2×C38).2C22, C2.3(C22×C38), SmallGroup(304,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C19
 Chief series C1 — C2 — C38 — C2×C38 — D4×C19 — C4○D4×C19
 Lower central C1 — C2 — C4○D4×C19
 Upper central C1 — C76 — C4○D4×C19

Generators and relations for C4○D4×C19
G = < a,b,c,d | a19=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Smallest permutation representation of C4○D4×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 56 119 81)(2 57 120 82)(3 39 121 83)(4 40 122 84)(5 41 123 85)(6 42 124 86)(7 43 125 87)(8 44 126 88)(9 45 127 89)(10 46 128 90)(11 47 129 91)(12 48 130 92)(13 49 131 93)(14 50 132 94)(15 51 133 95)(16 52 115 77)(17 53 116 78)(18 54 117 79)(19 55 118 80)(20 113 68 148)(21 114 69 149)(22 96 70 150)(23 97 71 151)(24 98 72 152)(25 99 73 134)(26 100 74 135)(27 101 75 136)(28 102 76 137)(29 103 58 138)(30 104 59 139)(31 105 60 140)(32 106 61 141)(33 107 62 142)(34 108 63 143)(35 109 64 144)(36 110 65 145)(37 111 66 146)(38 112 67 147)
(1 81 119 56)(2 82 120 57)(3 83 121 39)(4 84 122 40)(5 85 123 41)(6 86 124 42)(7 87 125 43)(8 88 126 44)(9 89 127 45)(10 90 128 46)(11 91 129 47)(12 92 130 48)(13 93 131 49)(14 94 132 50)(15 95 133 51)(16 77 115 52)(17 78 116 53)(18 79 117 54)(19 80 118 55)(20 113 68 148)(21 114 69 149)(22 96 70 150)(23 97 71 151)(24 98 72 152)(25 99 73 134)(26 100 74 135)(27 101 75 136)(28 102 76 137)(29 103 58 138)(30 104 59 139)(31 105 60 140)(32 106 61 141)(33 107 62 142)(34 108 63 143)(35 109 64 144)(36 110 65 145)(37 111 66 146)(38 112 67 147)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(39 96)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 128)(59 129)(60 130)(61 131)(62 132)(63 133)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 121)(71 122)(72 123)(73 124)(74 125)(75 126)(76 127)(77 144)(78 145)(79 146)(80 147)(81 148)(82 149)(83 150)(84 151)(85 152)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)

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

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

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

190 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 19A ··· 19R 38A ··· 38R 38S ··· 38BT 76A ··· 76AJ 76AK ··· 76CL order 1 2 2 2 2 4 4 4 4 4 19 ··· 19 38 ··· 38 38 ··· 38 76 ··· 76 76 ··· 76 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

190 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C19 C38 C38 C38 C4○D4 C4○D4×C19 kernel C4○D4×C19 C2×C76 D4×C19 Q8×C19 C4○D4 C2×C4 D4 Q8 C19 C1 # reps 1 3 3 1 18 54 54 18 2 36

Matrix representation of C4○D4×C19 in GL2(𝔽229) generated by

 104 0 0 104
,
 122 0 0 122
,
 107 0 146 122
,
 83 214 47 146
G:=sub<GL(2,GF(229))| [104,0,0,104],[122,0,0,122],[107,146,0,122],[83,47,214,146] >;

C4○D4×C19 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

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