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## G = C2×C104⋊C2order 416 = 25·13

### Direct product of C2 and C104⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C104⋊C2
 Chief series C1 — C13 — C26 — C52 — D52 — C2×D52 — C2×C104⋊C2
 Lower central C13 — C26 — C52 — C2×C104⋊C2
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×C104⋊C2
G = < a,b,c | a2=b104=c2=1, ab=ba, ac=ca, cbc=b51 >

Subgroups: 624 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C13, C2×C8, SD16 [×4], C2×D4, C2×Q8, D13 [×2], C26, C26 [×2], C2×SD16, Dic13 [×2], C52 [×2], D26 [×4], C2×C26, C104 [×2], Dic26 [×2], Dic26, D52 [×2], D52, C2×Dic13, C2×C52, C22×D13, C104⋊C2 [×4], C2×C104, C2×Dic26, C2×D52, C2×C104⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, SD16 [×2], C2×D4, D13, C2×SD16, D26 [×3], D52 [×2], C22×D13, C104⋊C2 [×2], C2×D52, C2×C104⋊C2

Smallest permutation representation of C2×C104⋊C2
On 208 points
Generators in S208
(1 132)(2 133)(3 134)(4 135)(5 136)(6 137)(7 138)(8 139)(9 140)(10 141)(11 142)(12 143)(13 144)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 151)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 161)(31 162)(32 163)(33 164)(34 165)(35 166)(36 167)(37 168)(38 169)(39 170)(40 171)(41 172)(42 173)(43 174)(44 175)(45 176)(46 177)(47 178)(48 179)(49 180)(50 181)(51 182)(52 183)(53 184)(54 185)(55 186)(56 187)(57 188)(58 189)(59 190)(60 191)(61 192)(62 193)(63 194)(64 195)(65 196)(66 197)(67 198)(68 199)(69 200)(70 201)(71 202)(72 203)(73 204)(74 205)(75 206)(76 207)(77 208)(78 105)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 128)(102 129)(103 130)(104 131)
(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)
(1 132)(2 183)(3 130)(4 181)(5 128)(6 179)(7 126)(8 177)(9 124)(10 175)(11 122)(12 173)(13 120)(14 171)(15 118)(16 169)(17 116)(18 167)(19 114)(20 165)(21 112)(22 163)(23 110)(24 161)(25 108)(26 159)(27 106)(28 157)(29 208)(30 155)(31 206)(32 153)(33 204)(34 151)(35 202)(36 149)(37 200)(38 147)(39 198)(40 145)(41 196)(42 143)(43 194)(44 141)(45 192)(46 139)(47 190)(48 137)(49 188)(50 135)(51 186)(52 133)(53 184)(54 131)(55 182)(56 129)(57 180)(58 127)(59 178)(60 125)(61 176)(62 123)(63 174)(64 121)(65 172)(66 119)(67 170)(68 117)(69 168)(70 115)(71 166)(72 113)(73 164)(74 111)(75 162)(76 109)(77 160)(78 107)(79 158)(80 105)(81 156)(82 207)(83 154)(84 205)(85 152)(86 203)(87 150)(88 201)(89 148)(90 199)(91 146)(92 197)(93 144)(94 195)(95 142)(96 193)(97 140)(98 191)(99 138)(100 189)(101 136)(102 187)(103 134)(104 185)

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

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

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

110 conjugacy classes

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

110 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 SD16 D13 D26 D26 D52 D52 C104⋊C2 kernel C2×C104⋊C2 C104⋊C2 C2×C104 C2×Dic26 C2×D52 C52 C2×C26 C26 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 4 6 12 6 12 12 48

Matrix representation of C2×C104⋊C2 in GL4(𝔽313) generated by

 312 0 0 0 0 312 0 0 0 0 1 0 0 0 0 1
,
 113 27 0 0 205 85 0 0 0 0 233 64 0 0 290 171
,
 110 162 0 0 105 203 0 0 0 0 241 6 0 0 23 72
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,1,0,0,0,0,1],[113,205,0,0,27,85,0,0,0,0,233,290,0,0,64,171],[110,105,0,0,162,203,0,0,0,0,241,23,0,0,6,72] >;

C2×C104⋊C2 in GAP, Magma, Sage, TeX

C_2\times C_{104}\rtimes C_2
% in TeX

G:=Group("C2xC104:C2");
// GroupNames label

G:=SmallGroup(416,123);
// by ID

G=gap.SmallGroup(416,123);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,218,50,579,69,13829]);
// 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^51>;
// generators/relations

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