Copied to
clipboard

## G = C37⋊D4order 296 = 23·37

### The semidirect product of C37 and D4 acting via D4/C22=C2

Aliases: C372D4, C22⋊D37, D742C2, Dic37⋊C2, C2.5D74, C74.5C22, (C2×C74)⋊2C2, SmallGroup(296,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C74 — C37⋊D4
 Chief series C1 — C37 — C74 — D74 — C37⋊D4
 Lower central C37 — C74 — C37⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C37⋊D4
G = < a,b,c | a37=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Smallest permutation representation of C37⋊D4
On 148 points
Generators in S148
(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)
(1 111 71 131)(2 110 72 130)(3 109 73 129)(4 108 74 128)(5 107 38 127)(6 106 39 126)(7 105 40 125)(8 104 41 124)(9 103 42 123)(10 102 43 122)(11 101 44 121)(12 100 45 120)(13 99 46 119)(14 98 47 118)(15 97 48 117)(16 96 49 116)(17 95 50 115)(18 94 51 114)(19 93 52 113)(20 92 53 112)(21 91 54 148)(22 90 55 147)(23 89 56 146)(24 88 57 145)(25 87 58 144)(26 86 59 143)(27 85 60 142)(28 84 61 141)(29 83 62 140)(30 82 63 139)(31 81 64 138)(32 80 65 137)(33 79 66 136)(34 78 67 135)(35 77 68 134)(36 76 69 133)(37 75 70 132)
(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 20)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(68 74)(69 73)(70 72)(75 130)(76 129)(77 128)(78 127)(79 126)(80 125)(81 124)(82 123)(83 122)(84 121)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)(91 114)(92 113)(93 112)(94 148)(95 147)(96 146)(97 145)(98 144)(99 143)(100 142)(101 141)(102 140)(103 139)(104 138)(105 137)(106 136)(107 135)(108 134)(109 133)(110 132)(111 131)

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

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

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

77 conjugacy classes

 class 1 2A 2B 2C 4 37A ··· 37R 74A ··· 74BB order 1 2 2 2 4 37 ··· 37 74 ··· 74 size 1 1 2 74 74 2 ··· 2 2 ··· 2

77 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D37 D74 C37⋊D4 kernel C37⋊D4 Dic37 D74 C2×C74 C37 C22 C2 C1 # reps 1 1 1 1 1 18 18 36

Matrix representation of C37⋊D4 in GL2(𝔽149) generated by

 84 1 73 47
,
 41 138 31 108
,
 47 18 76 102
G:=sub<GL(2,GF(149))| [84,73,1,47],[41,31,138,108],[47,76,18,102] >;

C37⋊D4 in GAP, Magma, Sage, TeX

C_{37}\rtimes D_4
% in TeX

G:=Group("C37:D4");
// GroupNames label

G:=SmallGroup(296,8);
// by ID

G=gap.SmallGroup(296,8);
# by ID

G:=PCGroup([4,-2,-2,-2,-37,49,4611]);
// Polycyclic

G:=Group<a,b,c|a^37=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

׿
×
𝔽