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G = C37⋊D4order 296 = 23·37

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

metabelian, supersoluble, monomial, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C74 — C37⋊D4
C1C37C74D74 — C37⋊D4
C37C74 — C37⋊D4
C1C2C22

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

2C2
74C2
37C4
37C22
2D37
2C74
37D4

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 2A2B2C 4 37A···37R74A···74BB
order1222437···3774···74
size11274742···22···2

77 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D37D74C37⋊D4
kernelC37⋊D4Dic37D74C2×C74C37C22C2C1
# reps11111181836

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

841
7347
,
41138
31108
,
4718
76102
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

Subgroup lattice of C37⋊D4 in TeX

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