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G = D4×C49order 392 = 23·72

Direct product of C49 and D4

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

Aliases: D4×C49, C4⋊C98, C22⋊C98, C1963C2, C28.3C14, C98.6C22, C7.(C7×D4), (C7×D4).C7, (C2×C98)⋊1C2, C2.1(C2×C98), (C2×C14).1C14, C14.6(C2×C14), SmallGroup(392,9)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C49
C1C7C14C98C2×C98 — D4×C49
C1C2 — D4×C49
C1C98 — D4×C49

Generators and relations for D4×C49
 G = < a,b,c | a49=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C14
2C14
2C98
2C98

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

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

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

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

245 conjugacy classes

class 1 2A2B2C 4 7A···7F14A···14F14G···14R28A···28F49A···49AP98A···98AP98AQ···98DV196A···196AP
order122247···714···1414···1428···2849···4998···9898···98196···196
size112221···11···12···22···21···11···12···22···2

245 irreducible representations

dim111111111222
type++++
imageC1C2C2C7C14C14C49C98C98D4C7×D4D4×C49
kernelD4×C49C196C2×C98C7×D4C28C2×C14D4C4C22C49C7C1
# reps11266124242841642

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

1710
0171
,
98172
6999
,
120
0196
G:=sub<GL(2,GF(197))| [171,0,0,171],[98,69,172,99],[1,0,20,196] >;

D4×C49 in GAP, Magma, Sage, TeX

D_4\times C_{49}
% in TeX

G:=Group("D4xC49");
// GroupNames label

G:=SmallGroup(392,9);
// by ID

G=gap.SmallGroup(392,9);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-7,301,222]);
// Polycyclic

G:=Group<a,b,c|a^49=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×C49 in TeX

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