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G = D4×C37order 296 = 23·37

Direct product of C37 and D4

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

Aliases: D4×C37, C4⋊C74, C22⋊C74, C1483C2, C74.6C22, (C2×C74)⋊1C2, C2.1(C2×C74), SmallGroup(296,10)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C37
C1C2C74C2×C74 — D4×C37
C1C2 — D4×C37
C1C74 — D4×C37

Generators and relations for D4×C37
 G = < a,b,c | a37=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C74
2C74

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

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

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

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

185 conjugacy classes

class 1 2A2B2C 4 37A···37AJ74A···74AJ74AK···74DD148A···148AJ
order1222437···3774···7474···74148···148
size112221···11···12···22···2

185 irreducible representations

dim11111122
type++++
imageC1C2C2C37C74C74D4D4×C37
kernelD4×C37C148C2×C74D4C4C22C37C1
# reps112363672136

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

1020
0102
,
1092
1940
,
10
40148
G:=sub<GL(2,GF(149))| [102,0,0,102],[109,19,2,40],[1,40,0,148] >;

D4×C37 in GAP, Magma, Sage, TeX

D_4\times C_{37}
% in TeX

G:=Group("D4xC37");
// GroupNames label

G:=SmallGroup(296,10);
// by ID

G=gap.SmallGroup(296,10);
# by ID

G:=PCGroup([4,-2,-2,-37,-2,1201]);
// Polycyclic

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

Export

Subgroup lattice of D4×C37 in TeX

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