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G = C4⋊D52order 416 = 25·13

The semidirect product of C4 and D52 acting via D52/C52=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C41D52, C524D4, C425D13, (C4×C52)⋊4C2, (C2×D52)⋊1C2, C26.3(C2×D4), C2.5(C2×D52), C131(C41D4), (C2×C4).76D26, (C2×C26).15C23, (C2×C52).87C22, (C22×D13).1C22, C22.36(C22×D13), SmallGroup(416,95)

Series: Derived Chief Lower central Upper central

C1C2×C26 — C4⋊D52
C1C13C26C2×C26C22×D13C2×D52 — C4⋊D52
C13C2×C26 — C4⋊D52
C1C22C42

Generators and relations for C4⋊D52
 G = < a,b,c | a4=b52=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 1080 in 108 conjugacy classes, 41 normal (7 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C22, C22 [×12], C2×C4 [×3], D4 [×12], C23 [×4], C13, C42, C2×D4 [×6], D13 [×4], C26 [×3], C41D4, C52 [×6], D26 [×12], C2×C26, D52 [×12], C2×C52 [×3], C22×D13 [×4], C4×C52, C2×D52 [×6], C4⋊D52
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], D13, C41D4, D26 [×3], D52 [×6], C22×D13, C2×D52 [×3], C4⋊D52

Smallest permutation representation of C4⋊D52
On 208 points
Generators in S208
(1 55 154 159)(2 56 155 160)(3 57 156 161)(4 58 105 162)(5 59 106 163)(6 60 107 164)(7 61 108 165)(8 62 109 166)(9 63 110 167)(10 64 111 168)(11 65 112 169)(12 66 113 170)(13 67 114 171)(14 68 115 172)(15 69 116 173)(16 70 117 174)(17 71 118 175)(18 72 119 176)(19 73 120 177)(20 74 121 178)(21 75 122 179)(22 76 123 180)(23 77 124 181)(24 78 125 182)(25 79 126 183)(26 80 127 184)(27 81 128 185)(28 82 129 186)(29 83 130 187)(30 84 131 188)(31 85 132 189)(32 86 133 190)(33 87 134 191)(34 88 135 192)(35 89 136 193)(36 90 137 194)(37 91 138 195)(38 92 139 196)(39 93 140 197)(40 94 141 198)(41 95 142 199)(42 96 143 200)(43 97 144 201)(44 98 145 202)(45 99 146 203)(46 100 147 204)(47 101 148 205)(48 102 149 206)(49 103 150 207)(50 104 151 208)(51 53 152 157)(52 54 153 158)
(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)
(1 197)(2 196)(3 195)(4 194)(5 193)(6 192)(7 191)(8 190)(9 189)(10 188)(11 187)(12 186)(13 185)(14 184)(15 183)(16 182)(17 181)(18 180)(19 179)(20 178)(21 177)(22 176)(23 175)(24 174)(25 173)(26 172)(27 171)(28 170)(29 169)(30 168)(31 167)(32 166)(33 165)(34 164)(35 163)(36 162)(37 161)(38 160)(39 159)(40 158)(41 157)(42 208)(43 207)(44 206)(45 205)(46 204)(47 203)(48 202)(49 201)(50 200)(51 199)(52 198)(53 142)(54 141)(55 140)(56 139)(57 138)(58 137)(59 136)(60 135)(61 134)(62 133)(63 132)(64 131)(65 130)(66 129)(67 128)(68 127)(69 126)(70 125)(71 124)(72 123)(73 122)(74 121)(75 120)(76 119)(77 118)(78 117)(79 116)(80 115)(81 114)(82 113)(83 112)(84 111)(85 110)(86 109)(87 108)(88 107)(89 106)(90 105)(91 156)(92 155)(93 154)(94 153)(95 152)(96 151)(97 150)(98 149)(99 148)(100 147)(101 146)(102 145)(103 144)(104 143)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F13A···13F26A···26R52A···52BT
order122222224···413···1326···2652···52
size1111525252522···22···22···22···2

110 irreducible representations

dim1112222
type+++++++
imageC1C2C2D4D13D26D52
kernelC4⋊D52C4×C52C2×D52C52C42C2×C4C4
# reps116661872

Matrix representation of C4⋊D52 in GL4(𝔽53) generated by

341600
371900
0010
0001
,
45100
23900
00451
00239
,
49200
19400
00926
00144
G:=sub<GL(4,GF(53))| [34,37,0,0,16,19,0,0,0,0,1,0,0,0,0,1],[4,2,0,0,51,39,0,0,0,0,4,2,0,0,51,39],[49,19,0,0,2,4,0,0,0,0,9,1,0,0,26,44] >;

C4⋊D52 in GAP, Magma, Sage, TeX

C_4\rtimes D_{52}
% in TeX

G:=Group("C4:D52");
// GroupNames label

G:=SmallGroup(416,95);
// by ID

G=gap.SmallGroup(416,95);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,218,50,13829]);
// Polycyclic

G:=Group<a,b,c|a^4=b^52=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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