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G = C42D52order 416 = 25·13

The semidirect product of C4 and D52 acting via D52/D26=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42D52, C521D4, D262D4, C4⋊C43D13, (C2×D52)⋊4C2, C2.9(C2×D52), C26.7(C2×D4), C132(C4⋊D4), (C2×C4).12D26, C2.13(D4×D13), D26⋊C48C2, (C2×C52).5C22, C26.34(C4○D4), (C2×C26).36C23, C2.6(D52⋊C2), (C22×D13).7C22, C22.50(C22×D13), (C2×Dic13).34C22, (C2×C4×D13)⋊1C2, (C13×C4⋊C4)⋊6C2, SmallGroup(416,116)

Series: Derived Chief Lower central Upper central

C1C2×C26 — C42D52
C1C13C26C2×C26C22×D13C2×C4×D13 — C42D52
C13C2×C26 — C42D52
C1C22C4⋊C4

Generators and relations for C42D52
 G = < a,b,c | a52=b4=c2=1, bab-1=a27, cac=a-1, cbc=b-1 >

Subgroups: 856 in 94 conjugacy classes, 35 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×10], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], C13, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], D13 [×4], C26 [×3], C4⋊D4, Dic13, C52 [×2], C52 [×2], D26 [×2], D26 [×8], C2×C26, C4×D13 [×2], D52 [×6], C2×Dic13, C2×C52, C2×C52 [×2], C22×D13, C22×D13 [×2], D26⋊C4 [×2], C13×C4⋊C4, C2×C4×D13, C2×D52, C2×D52 [×2], C42D52
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, D13, C4⋊D4, D26 [×3], D52 [×2], C22×D13, C2×D52, D4×D13, D52⋊C2, C42D52

Smallest permutation representation of C42D52
On 208 points
Generators in S208
(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)
(1 157 95 114)(2 184 96 141)(3 159 97 116)(4 186 98 143)(5 161 99 118)(6 188 100 145)(7 163 101 120)(8 190 102 147)(9 165 103 122)(10 192 104 149)(11 167 53 124)(12 194 54 151)(13 169 55 126)(14 196 56 153)(15 171 57 128)(16 198 58 155)(17 173 59 130)(18 200 60 105)(19 175 61 132)(20 202 62 107)(21 177 63 134)(22 204 64 109)(23 179 65 136)(24 206 66 111)(25 181 67 138)(26 208 68 113)(27 183 69 140)(28 158 70 115)(29 185 71 142)(30 160 72 117)(31 187 73 144)(32 162 74 119)(33 189 75 146)(34 164 76 121)(35 191 77 148)(36 166 78 123)(37 193 79 150)(38 168 80 125)(39 195 81 152)(40 170 82 127)(41 197 83 154)(42 172 84 129)(43 199 85 156)(44 174 86 131)(45 201 87 106)(46 176 88 133)(47 203 89 108)(48 178 90 135)(49 205 91 110)(50 180 92 137)(51 207 93 112)(52 182 94 139)
(1 114)(2 113)(3 112)(4 111)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 156)(12 155)(13 154)(14 153)(15 152)(16 151)(17 150)(18 149)(19 148)(20 147)(21 146)(22 145)(23 144)(24 143)(25 142)(26 141)(27 140)(28 139)(29 138)(30 137)(31 136)(32 135)(33 134)(34 133)(35 132)(36 131)(37 130)(38 129)(39 128)(40 127)(41 126)(42 125)(43 124)(44 123)(45 122)(46 121)(47 120)(48 119)(49 118)(50 117)(51 116)(52 115)(53 199)(54 198)(55 197)(56 196)(57 195)(58 194)(59 193)(60 192)(61 191)(62 190)(63 189)(64 188)(65 187)(66 186)(67 185)(68 184)(69 183)(70 182)(71 181)(72 180)(73 179)(74 178)(75 177)(76 176)(77 175)(78 174)(79 173)(80 172)(81 171)(82 170)(83 169)(84 168)(85 167)(86 166)(87 165)(88 164)(89 163)(90 162)(91 161)(92 160)(93 159)(94 158)(95 157)(96 208)(97 207)(98 206)(99 205)(100 204)(101 203)(102 202)(103 201)(104 200)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F13A···13F26A···26R52A···52AJ
order1222222244444413···1326···2652···52
size111126265252224426262···22···24···4

74 irreducible representations

dim1111122222244
type++++++++++++
imageC1C2C2C2C2D4D4C4○D4D13D26D52D4×D13D52⋊C2
kernelC42D52D26⋊C4C13×C4⋊C4C2×C4×D13C2×D52C52D26C26C4⋊C4C2×C4C4C2C2
# reps121132226182466

Matrix representation of C42D52 in GL4(𝔽53) generated by

484400
244300
00300
002923
,
32100
505000
003713
002116
,
2800
465100
003713
001316
G:=sub<GL(4,GF(53))| [48,24,0,0,44,43,0,0,0,0,30,29,0,0,0,23],[3,50,0,0,21,50,0,0,0,0,37,21,0,0,13,16],[2,46,0,0,8,51,0,0,0,0,37,13,0,0,13,16] >;

C42D52 in GAP, Magma, Sage, TeX

C_4\rtimes_2D_{52}
% in TeX

G:=Group("C4:2D52");
// GroupNames label

G:=SmallGroup(416,116);
// by ID

G=gap.SmallGroup(416,116);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,188,50,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^4=c^2=1,b*a*b^-1=a^27,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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