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## G = C2×D104order 416 = 25·13

### Direct product of C2 and D104

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×D104
 Chief series C1 — C13 — C26 — C52 — D52 — C2×D52 — C2×D104
 Lower central C13 — C26 — C52 — C2×D104
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×D104
G = < a,b,c | a2=b104=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 848 in 76 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C8 [×2], C2×C4, D4 [×6], C23 [×2], C13, C2×C8, D8 [×4], C2×D4 [×2], D13 [×4], C26, C26 [×2], C2×D8, C52 [×2], D26 [×8], C2×C26, C104 [×2], D52 [×4], D52 [×2], C2×C52, C22×D13 [×2], D104 [×4], C2×C104, C2×D52 [×2], C2×D104
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], C2×D4, D13, C2×D8, D26 [×3], D52 [×2], C22×D13, D104 [×2], C2×D52, C2×D104

Smallest permutation representation of C2×D104
On 208 points
Generators in S208
(1 128)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 136)(10 137)(11 138)(12 139)(13 140)(14 141)(15 142)(16 143)(17 144)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 151)(25 152)(26 153)(27 154)(28 155)(29 156)(30 157)(31 158)(32 159)(33 160)(34 161)(35 162)(36 163)(37 164)(38 165)(39 166)(40 167)(41 168)(42 169)(43 170)(44 171)(45 172)(46 173)(47 174)(48 175)(49 176)(50 177)(51 178)(52 179)(53 180)(54 181)(55 182)(56 183)(57 184)(58 185)(59 186)(60 187)(61 188)(62 189)(63 190)(64 191)(65 192)(66 193)(67 194)(68 195)(69 196)(70 197)(71 198)(72 199)(73 200)(74 201)(75 202)(76 203)(77 204)(78 205)(79 206)(80 207)(81 208)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)
(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)(105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 127)(2 126)(3 125)(4 124)(5 123)(6 122)(7 121)(8 120)(9 119)(10 118)(11 117)(12 116)(13 115)(14 114)(15 113)(16 112)(17 111)(18 110)(19 109)(20 108)(21 107)(22 106)(23 105)(24 208)(25 207)(26 206)(27 205)(28 204)(29 203)(30 202)(31 201)(32 200)(33 199)(34 198)(35 197)(36 196)(37 195)(38 194)(39 193)(40 192)(41 191)(42 190)(43 189)(44 188)(45 187)(46 186)(47 185)(48 184)(49 183)(50 182)(51 181)(52 180)(53 179)(54 178)(55 177)(56 176)(57 175)(58 174)(59 173)(60 172)(61 171)(62 170)(63 169)(64 168)(65 167)(66 166)(67 165)(68 164)(69 163)(70 162)(71 161)(72 160)(73 159)(74 158)(75 157)(76 156)(77 155)(78 154)(79 153)(80 152)(81 151)(82 150)(83 149)(84 148)(85 147)(86 146)(87 145)(88 144)(89 143)(90 142)(91 141)(92 140)(93 139)(94 138)(95 137)(96 136)(97 135)(98 134)(99 133)(100 132)(101 131)(102 130)(103 129)(104 128)

G:=sub<Sym(208)| (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,152)(26,153)(27,154)(28,155)(29,156)(30,157)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,164)(38,165)(39,166)(40,167)(41,168)(42,169)(43,170)(44,171)(45,172)(46,173)(47,174)(48,175)(49,176)(50,177)(51,178)(52,179)(53,180)(54,181)(55,182)(56,183)(57,184)(58,185)(59,186)(60,187)(61,188)(62,189)(63,190)(64,191)(65,192)(66,193)(67,194)(68,195)(69,196)(70,197)(71,198)(72,199)(73,200)(74,201)(75,202)(76,203)(77,204)(78,205)(79,206)(80,207)(81,208)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127), (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)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,120)(9,119)(10,118)(11,117)(12,116)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,208)(25,207)(26,206)(27,205)(28,204)(29,203)(30,202)(31,201)(32,200)(33,199)(34,198)(35,197)(36,196)(37,195)(38,194)(39,193)(40,192)(41,191)(42,190)(43,189)(44,188)(45,187)(46,186)(47,185)(48,184)(49,183)(50,182)(51,181)(52,180)(53,179)(54,178)(55,177)(56,176)(57,175)(58,174)(59,173)(60,172)(61,171)(62,170)(63,169)(64,168)(65,167)(66,166)(67,165)(68,164)(69,163)(70,162)(71,161)(72,160)(73,159)(74,158)(75,157)(76,156)(77,155)(78,154)(79,153)(80,152)(81,151)(82,150)(83,149)(84,148)(85,147)(86,146)(87,145)(88,144)(89,143)(90,142)(91,141)(92,140)(93,139)(94,138)(95,137)(96,136)(97,135)(98,134)(99,133)(100,132)(101,131)(102,130)(103,129)(104,128)>;

G:=Group( (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,152)(26,153)(27,154)(28,155)(29,156)(30,157)(31,158)(32,159)(33,160)(34,161)(35,162)(36,163)(37,164)(38,165)(39,166)(40,167)(41,168)(42,169)(43,170)(44,171)(45,172)(46,173)(47,174)(48,175)(49,176)(50,177)(51,178)(52,179)(53,180)(54,181)(55,182)(56,183)(57,184)(58,185)(59,186)(60,187)(61,188)(62,189)(63,190)(64,191)(65,192)(66,193)(67,194)(68,195)(69,196)(70,197)(71,198)(72,199)(73,200)(74,201)(75,202)(76,203)(77,204)(78,205)(79,206)(80,207)(81,208)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127), (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)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,127)(2,126)(3,125)(4,124)(5,123)(6,122)(7,121)(8,120)(9,119)(10,118)(11,117)(12,116)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,208)(25,207)(26,206)(27,205)(28,204)(29,203)(30,202)(31,201)(32,200)(33,199)(34,198)(35,197)(36,196)(37,195)(38,194)(39,193)(40,192)(41,191)(42,190)(43,189)(44,188)(45,187)(46,186)(47,185)(48,184)(49,183)(50,182)(51,181)(52,180)(53,179)(54,178)(55,177)(56,176)(57,175)(58,174)(59,173)(60,172)(61,171)(62,170)(63,169)(64,168)(65,167)(66,166)(67,165)(68,164)(69,163)(70,162)(71,161)(72,160)(73,159)(74,158)(75,157)(76,156)(77,155)(78,154)(79,153)(80,152)(81,151)(82,150)(83,149)(84,148)(85,147)(86,146)(87,145)(88,144)(89,143)(90,142)(91,141)(92,140)(93,139)(94,138)(95,137)(96,136)(97,135)(98,134)(99,133)(100,132)(101,131)(102,130)(103,129)(104,128) );

G=PermutationGroup([(1,128),(2,129),(3,130),(4,131),(5,132),(6,133),(7,134),(8,135),(9,136),(10,137),(11,138),(12,139),(13,140),(14,141),(15,142),(16,143),(17,144),(18,145),(19,146),(20,147),(21,148),(22,149),(23,150),(24,151),(25,152),(26,153),(27,154),(28,155),(29,156),(30,157),(31,158),(32,159),(33,160),(34,161),(35,162),(36,163),(37,164),(38,165),(39,166),(40,167),(41,168),(42,169),(43,170),(44,171),(45,172),(46,173),(47,174),(48,175),(49,176),(50,177),(51,178),(52,179),(53,180),(54,181),(55,182),(56,183),(57,184),(58,185),(59,186),(60,187),(61,188),(62,189),(63,190),(64,191),(65,192),(66,193),(67,194),(68,195),(69,196),(70,197),(71,198),(72,199),(73,200),(74,201),(75,202),(76,203),(77,204),(78,205),(79,206),(80,207),(81,208),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127)], [(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),(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,127),(2,126),(3,125),(4,124),(5,123),(6,122),(7,121),(8,120),(9,119),(10,118),(11,117),(12,116),(13,115),(14,114),(15,113),(16,112),(17,111),(18,110),(19,109),(20,108),(21,107),(22,106),(23,105),(24,208),(25,207),(26,206),(27,205),(28,204),(29,203),(30,202),(31,201),(32,200),(33,199),(34,198),(35,197),(36,196),(37,195),(38,194),(39,193),(40,192),(41,191),(42,190),(43,189),(44,188),(45,187),(46,186),(47,185),(48,184),(49,183),(50,182),(51,181),(52,180),(53,179),(54,178),(55,177),(56,176),(57,175),(58,174),(59,173),(60,172),(61,171),(62,170),(63,169),(64,168),(65,167),(66,166),(67,165),(68,164),(69,163),(70,162),(71,161),(72,160),(73,159),(74,158),(75,157),(76,156),(77,155),(78,154),(79,153),(80,152),(81,151),(82,150),(83,149),(84,148),(85,147),(86,146),(87,145),(88,144),(89,143),(90,142),(91,141),(92,140),(93,139),(94,138),(95,137),(96,136),(97,135),(98,134),(99,133),(100,132),(101,131),(102,130),(103,129),(104,128)])

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 8A 8B 8C 8D 13A ··· 13F 26A ··· 26R 52A ··· 52X 104A ··· 104AV order 1 2 2 2 2 2 2 2 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 1 1 52 52 52 52 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 D4 D4 D8 D13 D26 D26 D52 D52 D104 kernel C2×D104 D104 C2×C104 C2×D52 C52 C2×C26 C26 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 2 1 1 4 6 12 6 12 12 48

Matrix representation of C2×D104 in GL3(𝔽313) generated by

 312 0 0 0 312 0 0 0 312
,
 312 0 0 0 92 4 0 2 160
,
 312 0 0 0 65 163 0 241 248
G:=sub<GL(3,GF(313))| [312,0,0,0,312,0,0,0,312],[312,0,0,0,92,2,0,4,160],[312,0,0,0,65,241,0,163,248] >;

C2×D104 in GAP, Magma, Sage, TeX

C_2\times D_{104}
% in TeX

G:=Group("C2xD104");
// GroupNames label

G:=SmallGroup(416,124);
// by ID

G=gap.SmallGroup(416,124);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,122,579,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^2=b^104=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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