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G = C527D4order 416 = 25·13

1st semidirect product of C52 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C527D4, C221D52, C23.24D26, (C2×C26)⋊5D4, (C2×D52)⋊6C2, C523C49C2, C133(C4⋊D4), C43(C13⋊D4), (C22×C52)⋊6C2, (C2×C4).85D26, C26.43(C2×D4), C2.17(C2×D52), (C22×C4)⋊4D13, D26⋊C43C2, C26.19(C4○D4), (C2×C52).94C22, (C2×C26).48C23, C2.19(D525C2), (C22×C26).40C22, C22.56(C22×D13), (C2×Dic13).16C22, (C22×D13).10C22, (C2×C13⋊D4)⋊3C2, C2.7(C2×C13⋊D4), SmallGroup(416,151)

Series: Derived Chief Lower central Upper central

C1C2×C26 — C527D4
C1C13C26C2×C26C22×D13C2×D52 — C527D4
C13C2×C26 — C527D4
C1C22C22×C4

Generators and relations for C527D4
 G = < a,b,c | a52=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 712 in 94 conjugacy classes, 37 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×8], C2×C4 [×2], C2×C4 [×4], D4 [×6], C23, C23 [×2], C13, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], D13 [×2], C26 [×3], C26 [×2], C4⋊D4, Dic13 [×2], C52 [×2], C52, D26 [×6], C2×C26, C2×C26 [×2], C2×C26 [×2], D52 [×2], C2×Dic13 [×2], C13⋊D4 [×4], C2×C52 [×2], C2×C52 [×2], C22×D13 [×2], C22×C26, C523C4, D26⋊C4 [×2], C2×D52, C2×C13⋊D4 [×2], C22×C52, C527D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4, D13, C4⋊D4, D26 [×3], D52 [×2], C13⋊D4 [×2], C22×D13, C2×D52, D525C2, C2×C13⋊D4, C527D4

Smallest permutation representation of C527D4
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)
(1 172 137 94)(2 171 138 93)(3 170 139 92)(4 169 140 91)(5 168 141 90)(6 167 142 89)(7 166 143 88)(8 165 144 87)(9 164 145 86)(10 163 146 85)(11 162 147 84)(12 161 148 83)(13 160 149 82)(14 159 150 81)(15 158 151 80)(16 157 152 79)(17 208 153 78)(18 207 154 77)(19 206 155 76)(20 205 156 75)(21 204 105 74)(22 203 106 73)(23 202 107 72)(24 201 108 71)(25 200 109 70)(26 199 110 69)(27 198 111 68)(28 197 112 67)(29 196 113 66)(30 195 114 65)(31 194 115 64)(32 193 116 63)(33 192 117 62)(34 191 118 61)(35 190 119 60)(36 189 120 59)(37 188 121 58)(38 187 122 57)(39 186 123 56)(40 185 124 55)(41 184 125 54)(42 183 126 53)(43 182 127 104)(44 181 128 103)(45 180 129 102)(46 179 130 101)(47 178 131 100)(48 177 132 99)(49 176 133 98)(50 175 134 97)(51 174 135 96)(52 173 136 95)
(1 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 84)(12 83)(13 82)(14 81)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 72)(24 71)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 104)(44 103)(45 102)(46 101)(47 100)(48 99)(49 98)(50 97)(51 96)(52 95)(105 204)(106 203)(107 202)(108 201)(109 200)(110 199)(111 198)(112 197)(113 196)(114 195)(115 194)(116 193)(117 192)(118 191)(119 190)(120 189)(121 188)(122 187)(123 186)(124 185)(125 184)(126 183)(127 182)(128 181)(129 180)(130 179)(131 178)(132 177)(133 176)(134 175)(135 174)(136 173)(137 172)(138 171)(139 170)(140 169)(141 168)(142 167)(143 166)(144 165)(145 164)(146 163)(147 162)(148 161)(149 160)(150 159)(151 158)(152 157)(153 208)(154 207)(155 206)(156 205)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F13A···13F26A···26AP52A···52AV
order1222222244444413···1326···2652···52
size1111225252222252522···22···22···2

110 irreducible representations

dim111111222222222
type++++++++++++
imageC1C2C2C2C2C2D4D4C4○D4D13D26D26C13⋊D4D52D525C2
kernelC527D4C523C4D26⋊C4C2×D52C2×C13⋊D4C22×C52C52C2×C26C26C22×C4C2×C4C23C4C22C2
# reps1121212226126242424

Matrix representation of C527D4 in GL4(𝔽53) generated by

20000
0800
0090
0006
,
0100
52000
0001
00520
,
0100
1000
0001
0010
G:=sub<GL(4,GF(53))| [20,0,0,0,0,8,0,0,0,0,9,0,0,0,0,6],[0,52,0,0,1,0,0,0,0,0,0,52,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C527D4 in GAP, Magma, Sage, TeX

C_{52}\rtimes_7D_4
% in TeX

G:=Group("C52:7D4");
// GroupNames label

G:=SmallGroup(416,151);
// by ID

G=gap.SmallGroup(416,151);
# by ID

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

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

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