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G = C52.48D4order 416 = 25·13

5th non-split extension by C52 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52.48D4, C222Dic26, C23.20D26, (C2×C26)⋊3Q8, C523C48C2, C26.8(C2×Q8), C26.39(C2×D4), (C2×C4).83D26, C134(C22⋊Q8), (C2×Dic26)⋊6C2, C26.D42C2, (C22×C52).6C2, C2.9(C2×Dic26), (C22×C4).5D13, C26.15(C4○D4), C4.23(C13⋊D4), (C2×C52).91C22, (C2×C26).42C23, C23.D13.4C2, C2.17(D525C2), (C22×C26).34C22, C22.54(C22×D13), (C2×Dic13).14C22, C2.5(C2×C13⋊D4), SmallGroup(416,145)

Series: Derived Chief Lower central Upper central

C1C2×C26 — C52.48D4
C1C13C26C2×C26C2×Dic13C2×Dic26 — C52.48D4
C13C2×C26 — C52.48D4
C1C22C22×C4

Generators and relations for C52.48D4
 G = < a,b,c | a52=b4=1, c2=a26, bab-1=cac-1=a-1, cbc-1=a26b-1 >

Subgroups: 376 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C13, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C26 [×3], C26 [×2], C22⋊Q8, Dic13 [×4], C52 [×2], C52, C2×C26, C2×C26 [×2], C2×C26 [×2], Dic26 [×2], C2×Dic13 [×4], C2×C52 [×2], C2×C52 [×2], C22×C26, C26.D4 [×2], C523C4, C23.D13 [×2], C2×Dic26, C22×C52, C52.48D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4, D13, C22⋊Q8, D26 [×3], Dic26 [×2], C13⋊D4 [×2], C22×D13, C2×Dic26, D525C2, C2×C13⋊D4, C52.48D4

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

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H13A···13F26A···26AP52A···52AV
order1222224444444413···1326···2652···52
size1111222222525252522···22···22···2

110 irreducible representations

dim111111222222222
type+++++++-+++-
imageC1C2C2C2C2C2D4Q8C4○D4D13D26D26C13⋊D4Dic26D525C2
kernelC52.48D4C26.D4C523C4C23.D13C2×Dic26C22×C52C52C2×C26C26C22×C4C2×C4C23C4C22C2
# reps1212112226126242424

Matrix representation of C52.48D4 in GL4(𝔽53) generated by

33000
334500
0060
00529
,
303500
472300
004912
0034
,
303500
02300
004912
00124
G:=sub<GL(4,GF(53))| [33,33,0,0,0,45,0,0,0,0,6,52,0,0,0,9],[30,47,0,0,35,23,0,0,0,0,49,3,0,0,12,4],[30,0,0,0,35,23,0,0,0,0,49,12,0,0,12,4] >;

C52.48D4 in GAP, Magma, Sage, TeX

C_{52}._{48}D_4
% in TeX

G:=Group("C52.48D4");
// GroupNames label

G:=SmallGroup(416,145);
// by ID

G=gap.SmallGroup(416,145);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,217,103,218,13829]);
// Polycyclic

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

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