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G = C52.17D4order 416 = 25·13

17th non-split extension by C52 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52.17D4, C23.7D26, (C2×D4).6D13, (D4×C26).5C2, (C2×C4).50D26, C26.48(C2×D4), (C4×Dic13)⋊5C2, C133(C4.4D4), C4.7(C13⋊D4), C23.D139C2, (C2×Dic26)⋊10C2, C26.30(C4○D4), (C2×C52).33C22, (C2×C26).51C23, C2.16(D42D13), (C22×C26).19C22, C22.58(C22×D13), (C2×Dic13).18C22, C2.12(C2×C13⋊D4), SmallGroup(416,157)

Series: Derived Chief Lower central Upper central

C1C2×C26 — C52.17D4
C1C13C26C2×C26C2×Dic13C4×Dic13 — C52.17D4
C13C2×C26 — C52.17D4
C1C22C2×D4

Generators and relations for C52.17D4
 G = < a,b,c | a52=b4=1, c2=a26, bab-1=a25, cac-1=a-1, cbc-1=a26b-1 >

Subgroups: 392 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C13, C42, C22⋊C4 [×4], C2×D4, C2×Q8, C26, C26 [×2], C26 [×2], C4.4D4, Dic13 [×4], C52 [×2], C2×C26, C2×C26 [×6], Dic26 [×2], C2×Dic13 [×4], C2×C52, D4×C13 [×2], C22×C26 [×2], C4×Dic13, C23.D13 [×4], C2×Dic26, D4×C26, C52.17D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D4 [×2], D13, C4.4D4, D26 [×3], C13⋊D4 [×2], C22×D13, D42D13 [×2], C2×C13⋊D4, C52.17D4

Smallest permutation representation of C52.17D4
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)
(1 196 63 113)(2 169 64 138)(3 194 65 111)(4 167 66 136)(5 192 67 109)(6 165 68 134)(7 190 69 107)(8 163 70 132)(9 188 71 105)(10 161 72 130)(11 186 73 155)(12 159 74 128)(13 184 75 153)(14 157 76 126)(15 182 77 151)(16 207 78 124)(17 180 79 149)(18 205 80 122)(19 178 81 147)(20 203 82 120)(21 176 83 145)(22 201 84 118)(23 174 85 143)(24 199 86 116)(25 172 87 141)(26 197 88 114)(27 170 89 139)(28 195 90 112)(29 168 91 137)(30 193 92 110)(31 166 93 135)(32 191 94 108)(33 164 95 133)(34 189 96 106)(35 162 97 131)(36 187 98 156)(37 160 99 129)(38 185 100 154)(39 158 101 127)(40 183 102 152)(41 208 103 125)(42 181 104 150)(43 206 53 123)(44 179 54 148)(45 204 55 121)(46 177 56 146)(47 202 57 119)(48 175 58 144)(49 200 59 117)(50 173 60 142)(51 198 61 115)(52 171 62 140)
(1 183 27 157)(2 182 28 208)(3 181 29 207)(4 180 30 206)(5 179 31 205)(6 178 32 204)(7 177 33 203)(8 176 34 202)(9 175 35 201)(10 174 36 200)(11 173 37 199)(12 172 38 198)(13 171 39 197)(14 170 40 196)(15 169 41 195)(16 168 42 194)(17 167 43 193)(18 166 44 192)(19 165 45 191)(20 164 46 190)(21 163 47 189)(22 162 48 188)(23 161 49 187)(24 160 50 186)(25 159 51 185)(26 158 52 184)(53 110 79 136)(54 109 80 135)(55 108 81 134)(56 107 82 133)(57 106 83 132)(58 105 84 131)(59 156 85 130)(60 155 86 129)(61 154 87 128)(62 153 88 127)(63 152 89 126)(64 151 90 125)(65 150 91 124)(66 149 92 123)(67 148 93 122)(68 147 94 121)(69 146 95 120)(70 145 96 119)(71 144 97 118)(72 143 98 117)(73 142 99 116)(74 141 100 115)(75 140 101 114)(76 139 102 113)(77 138 103 112)(78 137 104 111)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H13A···13F26A···26R26S···26AP52A···52L
order1222224444444413···1326···2626···2652···52
size111144222626262652522···22···24···44···4

74 irreducible representations

dim111112222224
type+++++++++-
imageC1C2C2C2C2D4C4○D4D13D26D26C13⋊D4D42D13
kernelC52.17D4C4×Dic13C23.D13C2×Dic26D4×C26C52C26C2×D4C2×C4C23C4C2
# reps114112466122412

Matrix representation of C52.17D4 in GL4(𝔽53) generated by

37000
04300
005251
0011
,
0100
52000
00300
00030
,
0100
1000
002346
00030
G:=sub<GL(4,GF(53))| [37,0,0,0,0,43,0,0,0,0,52,1,0,0,51,1],[0,52,0,0,1,0,0,0,0,0,30,0,0,0,0,30],[0,1,0,0,1,0,0,0,0,0,23,0,0,0,46,30] >;

C52.17D4 in GAP, Magma, Sage, TeX

C_{52}._{17}D_4
% in TeX

G:=Group("C52.17D4");
// GroupNames label

G:=SmallGroup(416,157);
// by ID

G=gap.SmallGroup(416,157);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,55,506,116,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^4=1,c^2=a^26,b*a*b^-1=a^25,c*a*c^-1=a^-1,c*b*c^-1=a^26*b^-1>;
// generators/relations

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