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G = C4×D49order 392 = 23·72

Direct product of C4 and D49

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

Aliases: C4×D49, D98.C2, C1962C2, C28.5D7, C2.1D98, C14.7D14, Dic492C2, C98.2C22, C7.(C4×D7), C491(C2×C4), SmallGroup(392,4)

Series: Derived Chief Lower central Upper central

C1C49 — C4×D49
C1C7C49C98D98 — C4×D49
C49 — C4×D49
C1C4

Generators and relations for C4×D49
 G = < a,b,c | a4=b49=c2=1, ab=ba, ac=ca, cbc=b-1 >

49C2
49C2
49C22
49C4
7D7
7D7
49C2×C4
7D14
7Dic7
7C4×D7

Smallest permutation representation of C4×D49
On 196 points
Generators in S196
(1 151 52 108)(2 152 53 109)(3 153 54 110)(4 154 55 111)(5 155 56 112)(6 156 57 113)(7 157 58 114)(8 158 59 115)(9 159 60 116)(10 160 61 117)(11 161 62 118)(12 162 63 119)(13 163 64 120)(14 164 65 121)(15 165 66 122)(16 166 67 123)(17 167 68 124)(18 168 69 125)(19 169 70 126)(20 170 71 127)(21 171 72 128)(22 172 73 129)(23 173 74 130)(24 174 75 131)(25 175 76 132)(26 176 77 133)(27 177 78 134)(28 178 79 135)(29 179 80 136)(30 180 81 137)(31 181 82 138)(32 182 83 139)(33 183 84 140)(34 184 85 141)(35 185 86 142)(36 186 87 143)(37 187 88 144)(38 188 89 145)(39 189 90 146)(40 190 91 147)(41 191 92 99)(42 192 93 100)(43 193 94 101)(44 194 95 102)(45 195 96 103)(46 196 97 104)(47 148 98 105)(48 149 50 106)(49 150 51 107)
(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)(148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)
(1 51)(2 50)(3 98)(4 97)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 90)(12 89)(13 88)(14 87)(15 86)(16 85)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 76)(26 75)(27 74)(28 73)(29 72)(30 71)(31 70)(32 69)(33 68)(34 67)(35 66)(36 65)(37 64)(38 63)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(99 159)(100 158)(101 157)(102 156)(103 155)(104 154)(105 153)(106 152)(107 151)(108 150)(109 149)(110 148)(111 196)(112 195)(113 194)(114 193)(115 192)(116 191)(117 190)(118 189)(119 188)(120 187)(121 186)(122 185)(123 184)(124 183)(125 182)(126 181)(127 180)(128 179)(129 178)(130 177)(131 176)(132 175)(133 174)(134 173)(135 172)(136 171)(137 170)(138 169)(139 168)(140 167)(141 166)(142 165)(143 164)(144 163)(145 162)(146 161)(147 160)

G:=sub<Sym(196)| (1,151,52,108)(2,152,53,109)(3,153,54,110)(4,154,55,111)(5,155,56,112)(6,156,57,113)(7,157,58,114)(8,158,59,115)(9,159,60,116)(10,160,61,117)(11,161,62,118)(12,162,63,119)(13,163,64,120)(14,164,65,121)(15,165,66,122)(16,166,67,123)(17,167,68,124)(18,168,69,125)(19,169,70,126)(20,170,71,127)(21,171,72,128)(22,172,73,129)(23,173,74,130)(24,174,75,131)(25,175,76,132)(26,176,77,133)(27,177,78,134)(28,178,79,135)(29,179,80,136)(30,180,81,137)(31,181,82,138)(32,182,83,139)(33,183,84,140)(34,184,85,141)(35,185,86,142)(36,186,87,143)(37,187,88,144)(38,188,89,145)(39,189,90,146)(40,190,91,147)(41,191,92,99)(42,192,93,100)(43,193,94,101)(44,194,95,102)(45,195,96,103)(46,196,97,104)(47,148,98,105)(48,149,50,106)(49,150,51,107), (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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196), (1,51)(2,50)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(99,159)(100,158)(101,157)(102,156)(103,155)(104,154)(105,153)(106,152)(107,151)(108,150)(109,149)(110,148)(111,196)(112,195)(113,194)(114,193)(115,192)(116,191)(117,190)(118,189)(119,188)(120,187)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,180)(128,179)(129,178)(130,177)(131,176)(132,175)(133,174)(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160)>;

G:=Group( (1,151,52,108)(2,152,53,109)(3,153,54,110)(4,154,55,111)(5,155,56,112)(6,156,57,113)(7,157,58,114)(8,158,59,115)(9,159,60,116)(10,160,61,117)(11,161,62,118)(12,162,63,119)(13,163,64,120)(14,164,65,121)(15,165,66,122)(16,166,67,123)(17,167,68,124)(18,168,69,125)(19,169,70,126)(20,170,71,127)(21,171,72,128)(22,172,73,129)(23,173,74,130)(24,174,75,131)(25,175,76,132)(26,176,77,133)(27,177,78,134)(28,178,79,135)(29,179,80,136)(30,180,81,137)(31,181,82,138)(32,182,83,139)(33,183,84,140)(34,184,85,141)(35,185,86,142)(36,186,87,143)(37,187,88,144)(38,188,89,145)(39,189,90,146)(40,190,91,147)(41,191,92,99)(42,192,93,100)(43,193,94,101)(44,194,95,102)(45,195,96,103)(46,196,97,104)(47,148,98,105)(48,149,50,106)(49,150,51,107), (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)(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196), (1,51)(2,50)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(99,159)(100,158)(101,157)(102,156)(103,155)(104,154)(105,153)(106,152)(107,151)(108,150)(109,149)(110,148)(111,196)(112,195)(113,194)(114,193)(115,192)(116,191)(117,190)(118,189)(119,188)(120,187)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,180)(128,179)(129,178)(130,177)(131,176)(132,175)(133,174)(134,173)(135,172)(136,171)(137,170)(138,169)(139,168)(140,167)(141,166)(142,165)(143,164)(144,163)(145,162)(146,161)(147,160) );

G=PermutationGroup([(1,151,52,108),(2,152,53,109),(3,153,54,110),(4,154,55,111),(5,155,56,112),(6,156,57,113),(7,157,58,114),(8,158,59,115),(9,159,60,116),(10,160,61,117),(11,161,62,118),(12,162,63,119),(13,163,64,120),(14,164,65,121),(15,165,66,122),(16,166,67,123),(17,167,68,124),(18,168,69,125),(19,169,70,126),(20,170,71,127),(21,171,72,128),(22,172,73,129),(23,173,74,130),(24,174,75,131),(25,175,76,132),(26,176,77,133),(27,177,78,134),(28,178,79,135),(29,179,80,136),(30,180,81,137),(31,181,82,138),(32,182,83,139),(33,183,84,140),(34,184,85,141),(35,185,86,142),(36,186,87,143),(37,187,88,144),(38,188,89,145),(39,189,90,146),(40,190,91,147),(41,191,92,99),(42,192,93,100),(43,193,94,101),(44,194,95,102),(45,195,96,103),(46,196,97,104),(47,148,98,105),(48,149,50,106),(49,150,51,107)], [(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),(148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)], [(1,51),(2,50),(3,98),(4,97),(5,96),(6,95),(7,94),(8,93),(9,92),(10,91),(11,90),(12,89),(13,88),(14,87),(15,86),(16,85),(17,84),(18,83),(19,82),(20,81),(21,80),(22,79),(23,78),(24,77),(25,76),(26,75),(27,74),(28,73),(29,72),(30,71),(31,70),(32,69),(33,68),(34,67),(35,66),(36,65),(37,64),(38,63),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(99,159),(100,158),(101,157),(102,156),(103,155),(104,154),(105,153),(106,152),(107,151),(108,150),(109,149),(110,148),(111,196),(112,195),(113,194),(114,193),(115,192),(116,191),(117,190),(118,189),(119,188),(120,187),(121,186),(122,185),(123,184),(124,183),(125,182),(126,181),(127,180),(128,179),(129,178),(130,177),(131,176),(132,175),(133,174),(134,173),(135,172),(136,171),(137,170),(138,169),(139,168),(140,167),(141,166),(142,165),(143,164),(144,163),(145,162),(146,161),(147,160)])

104 conjugacy classes

class 1 2A2B2C4A4B4C4D7A7B7C14A14B14C28A···28F49A···49U98A···98U196A···196AP
order1222444477714141428···2849···4998···98196···196
size1149491149492222222···22···22···22···2

104 irreducible representations

dim11111222222
type++++++++
imageC1C2C2C2C4D7D14C4×D7D49D98C4×D49
kernelC4×D49Dic49C196D98D49C28C14C7C4C2C1
# reps11114336212142

Matrix representation of C4×D49 in GL2(𝔽197) generated by

1830
0183
,
12654
2898
,
10917
6688
G:=sub<GL(2,GF(197))| [183,0,0,183],[126,28,54,98],[109,66,17,88] >;

C4×D49 in GAP, Magma, Sage, TeX

C_4\times D_{49}
% in TeX

G:=Group("C4xD49");
// GroupNames label

G:=SmallGroup(392,4);
// by ID

G=gap.SmallGroup(392,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-7,26,2083,858,8404]);
// Polycyclic

G:=Group<a,b,c|a^4=b^49=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D49 in TeX

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