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## G = D4×C38order 304 = 24·19

### Direct product of C38 and D4

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

Aliases: D4×C38, C23⋊C38, C764C22, C38.11C23, C4⋊(C2×C38), (C2×C4)⋊2C38, (C2×C76)⋊6C2, C22⋊(C2×C38), (C22×C38)⋊1C2, (C2×C38)⋊2C22, C2.1(C22×C38), SmallGroup(304,38)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×C38
G = < a,b,c | a38=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C2×D4, C19, C38, C38, C38, C76, C2×C38, C2×C38, C2×C38, C2×C76, D4×C19, C22×C38, D4×C38
Quotients: C1, C2, C22, D4, C23, C2×D4, C19, C38, C2×C38, D4×C19, C22×C38, D4×C38

Smallest permutation representation of D4×C38
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 73 112 123)(2 74 113 124)(3 75 114 125)(4 76 77 126)(5 39 78 127)(6 40 79 128)(7 41 80 129)(8 42 81 130)(9 43 82 131)(10 44 83 132)(11 45 84 133)(12 46 85 134)(13 47 86 135)(14 48 87 136)(15 49 88 137)(16 50 89 138)(17 51 90 139)(18 52 91 140)(19 53 92 141)(20 54 93 142)(21 55 94 143)(22 56 95 144)(23 57 96 145)(24 58 97 146)(25 59 98 147)(26 60 99 148)(27 61 100 149)(28 62 101 150)(29 63 102 151)(30 64 103 152)(31 65 104 115)(32 66 105 116)(33 67 106 117)(34 68 107 118)(35 69 108 119)(36 70 109 120)(37 71 110 121)(38 72 111 122)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(77 145)(78 146)(79 147)(80 148)(81 149)(82 150)(83 151)(84 152)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 131)(102 132)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 139)(110 140)(111 141)(112 142)(113 143)(114 144)

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

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

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

190 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 19A ··· 19R 38A ··· 38BB 38BC ··· 38DV 76A ··· 76AJ order 1 2 2 2 2 2 2 2 4 4 19 ··· 19 38 ··· 38 38 ··· 38 76 ··· 76 size 1 1 1 1 2 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 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 D4 D4×C19 kernel D4×C38 C2×C76 D4×C19 C22×C38 C2×D4 C2×C4 D4 C23 C38 C2 # reps 1 1 4 2 18 18 72 36 2 36

Matrix representation of D4×C38 in GL3(𝔽229) generated by

 228 0 0 0 11 0 0 0 11
,
 1 0 0 0 0 1 0 228 0
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(229))| [228,0,0,0,11,0,0,0,11],[1,0,0,0,0,228,0,1,0],[1,0,0,0,0,1,0,1,0] >;

D4×C38 in GAP, Magma, Sage, TeX

D_4\times C_{38}
% in TeX

G:=Group("D4xC38");
// GroupNames label

G:=SmallGroup(304,38);
// by ID

G=gap.SmallGroup(304,38);
# by ID

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

G:=Group<a,b,c|a^38=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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