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G = C4×D37order 296 = 23·37

Direct product of C4 and D37

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D37, C1482C2, C2.1D74, D74.2C2, Dic372C2, C74.2C22, C372(C2×C4), SmallGroup(296,5)

Series: Derived Chief Lower central Upper central

C1C37 — C4×D37
C1C37C74D74 — C4×D37
C37 — C4×D37
C1C4

Generators and relations for C4×D37
 G = < a,b,c | a4=b37=c2=1, ab=ba, ac=ca, cbc=b-1 >

37C2
37C2
37C22
37C4
37C2×C4

Smallest permutation representation of C4×D37
On 148 points
Generators in S148
(1 121 65 103)(2 122 66 104)(3 123 67 105)(4 124 68 106)(5 125 69 107)(6 126 70 108)(7 127 71 109)(8 128 72 110)(9 129 73 111)(10 130 74 75)(11 131 38 76)(12 132 39 77)(13 133 40 78)(14 134 41 79)(15 135 42 80)(16 136 43 81)(17 137 44 82)(18 138 45 83)(19 139 46 84)(20 140 47 85)(21 141 48 86)(22 142 49 87)(23 143 50 88)(24 144 51 89)(25 145 52 90)(26 146 53 91)(27 147 54 92)(28 148 55 93)(29 112 56 94)(30 113 57 95)(31 114 58 96)(32 115 59 97)(33 116 60 98)(34 117 61 99)(35 118 62 100)(36 119 63 101)(37 120 64 102)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(75 93)(76 92)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(112 129)(113 128)(114 127)(115 126)(116 125)(117 124)(118 123)(119 122)(120 121)(130 148)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 140)

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

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

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

80 conjugacy classes

class 1 2A2B2C4A4B4C4D37A···37R74A···74R148A···148AJ
order1222444437···3774···74148···148
size1137371137372···22···22···2

80 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D37D74C4×D37
kernelC4×D37Dic37C148D74D37C4C2C1
# reps11114181836

Matrix representation of C4×D37 in GL2(𝔽149) generated by

440
044
,
01
148122
,
01
10
G:=sub<GL(2,GF(149))| [44,0,0,44],[0,148,1,122],[0,1,1,0] >;

C4×D37 in GAP, Magma, Sage, TeX

C_4\times D_{37}
% in TeX

G:=Group("C4xD37");
// GroupNames label

G:=SmallGroup(296,5);
// by ID

G=gap.SmallGroup(296,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-37,21,4611]);
// Polycyclic

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

Export

Subgroup lattice of C4×D37 in TeX

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