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G = C49⋊D4order 392 = 23·72

The semidirect product of C49 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C492D4, C22⋊D49, D982C2, Dic49⋊C2, C2.5D98, C14.10D14, C98.5C22, (C2×C98)⋊2C2, C7.(C7⋊D4), (C2×C14).2D7, SmallGroup(392,7)

Series: Derived Chief Lower central Upper central

C1C98 — C49⋊D4
C1C7C49C98D98 — C49⋊D4
C49C98 — C49⋊D4
C1C2C22

Generators and relations for C49⋊D4
 G = < a,b,c | a49=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
98C2
49C4
49C22
2C14
14D7
49D4
7Dic7
7D14
2D49
2C98
7C7⋊D4

Smallest permutation representation of C49⋊D4
On 196 points
Generators in S196
(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)
(1 108 80 162)(2 107 81 161)(3 106 82 160)(4 105 83 159)(5 104 84 158)(6 103 85 157)(7 102 86 156)(8 101 87 155)(9 100 88 154)(10 99 89 153)(11 147 90 152)(12 146 91 151)(13 145 92 150)(14 144 93 149)(15 143 94 148)(16 142 95 196)(17 141 96 195)(18 140 97 194)(19 139 98 193)(20 138 50 192)(21 137 51 191)(22 136 52 190)(23 135 53 189)(24 134 54 188)(25 133 55 187)(26 132 56 186)(27 131 57 185)(28 130 58 184)(29 129 59 183)(30 128 60 182)(31 127 61 181)(32 126 62 180)(33 125 63 179)(34 124 64 178)(35 123 65 177)(36 122 66 176)(37 121 67 175)(38 120 68 174)(39 119 69 173)(40 118 70 172)(41 117 71 171)(42 116 72 170)(43 115 73 169)(44 114 74 168)(45 113 75 167)(46 112 76 166)(47 111 77 165)(48 110 78 164)(49 109 79 163)
(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)(79 81)(99 171)(100 170)(101 169)(102 168)(103 167)(104 166)(105 165)(106 164)(107 163)(108 162)(109 161)(110 160)(111 159)(112 158)(113 157)(114 156)(115 155)(116 154)(117 153)(118 152)(119 151)(120 150)(121 149)(122 148)(123 196)(124 195)(125 194)(126 193)(127 192)(128 191)(129 190)(130 189)(131 188)(132 187)(133 186)(134 185)(135 184)(136 183)(137 182)(138 181)(139 180)(140 179)(141 178)(142 177)(143 176)(144 175)(145 174)(146 173)(147 172)

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

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

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

101 conjugacy classes

class 1 2A2B2C 4 7A7B7C14A···14I49A···49U98A···98BK
order1222477714···1449···4998···98
size11298982222···22···22···2

101 irreducible representations

dim11112222222
type+++++++++
imageC1C2C2C2D4D7D14C7⋊D4D49D98C49⋊D4
kernelC49⋊D4Dic49D98C2×C98C49C2×C14C14C7C22C2C1
# reps11111336212142

Matrix representation of C49⋊D4 in GL2(𝔽197) generated by

161169
2871
,
89146
24108
,
10
158196
G:=sub<GL(2,GF(197))| [161,28,169,71],[89,24,146,108],[1,158,0,196] >;

C49⋊D4 in GAP, Magma, Sage, TeX

C_{49}\rtimes D_4
% in TeX

G:=Group("C49:D4");
// GroupNames label

G:=SmallGroup(392,7);
// by ID

G=gap.SmallGroup(392,7);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-7,61,2083,858,8404]);
// Polycyclic

G:=Group<a,b,c|a^49=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 C49⋊D4 in TeX

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