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## G = C52⋊7D4order 416 = 25·13

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26 — C52⋊7D4
 Chief series C1 — C13 — C26 — C2×C26 — C22×D13 — C2×D52 — C52⋊7D4
 Lower central C13 — C2×C26 — C52⋊7D4
 Upper central C1 — C22 — C22×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, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C13, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D13, C26, C26, C4⋊D4, Dic13, C52, C52, D26, C2×C26, C2×C26, C2×C26, D52, C2×Dic13, C13⋊D4, C2×C52, C2×C52, C22×D13, C22×C26, C523C4, D26⋊C4, C2×D52, C2×C13⋊D4, C22×C52, C527D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, D13, C4⋊D4, D26, D52, C13⋊D4, 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)
(1 157 109 73)(2 208 110 72)(3 207 111 71)(4 206 112 70)(5 205 113 69)(6 204 114 68)(7 203 115 67)(8 202 116 66)(9 201 117 65)(10 200 118 64)(11 199 119 63)(12 198 120 62)(13 197 121 61)(14 196 122 60)(15 195 123 59)(16 194 124 58)(17 193 125 57)(18 192 126 56)(19 191 127 55)(20 190 128 54)(21 189 129 53)(22 188 130 104)(23 187 131 103)(24 186 132 102)(25 185 133 101)(26 184 134 100)(27 183 135 99)(28 182 136 98)(29 181 137 97)(30 180 138 96)(31 179 139 95)(32 178 140 94)(33 177 141 93)(34 176 142 92)(35 175 143 91)(36 174 144 90)(37 173 145 89)(38 172 146 88)(39 171 147 87)(40 170 148 86)(41 169 149 85)(42 168 150 84)(43 167 151 83)(44 166 152 82)(45 165 153 81)(46 164 154 80)(47 163 155 79)(48 162 156 78)(49 161 105 77)(50 160 106 76)(51 159 107 75)(52 158 108 74)
(1 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 104)(23 103)(24 102)(25 101)(26 100)(27 99)(28 98)(29 97)(30 96)(31 95)(32 94)(33 93)(34 92)(35 91)(36 90)(37 89)(38 88)(39 87)(40 86)(41 85)(42 84)(43 83)(44 82)(45 81)(46 80)(47 79)(48 78)(49 77)(50 76)(51 75)(52 74)(105 161)(106 160)(107 159)(108 158)(109 157)(110 208)(111 207)(112 206)(113 205)(114 204)(115 203)(116 202)(117 201)(118 200)(119 199)(120 198)(121 197)(122 196)(123 195)(124 194)(125 193)(126 192)(127 191)(128 190)(129 189)(130 188)(131 187)(132 186)(133 185)(134 184)(135 183)(136 182)(137 181)(138 180)(139 179)(140 178)(141 177)(142 176)(143 175)(144 174)(145 173)(146 172)(147 171)(148 170)(149 169)(150 168)(151 167)(152 166)(153 165)(154 164)(155 163)(156 162)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 13A ··· 13F 26A ··· 26AP 52A ··· 52AV order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 1 1 2 2 52 52 2 2 2 2 52 52 2 ··· 2 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 C4○D4 D13 D26 D26 C13⋊D4 D52 D52⋊5C2 kernel C52⋊7D4 C52⋊3C4 D26⋊C4 C2×D52 C2×C13⋊D4 C22×C52 C52 C2×C26 C26 C22×C4 C2×C4 C23 C4 C22 C2 # reps 1 1 2 1 2 1 2 2 2 6 12 6 24 24 24

Matrix representation of C527D4 in GL4(𝔽53) generated by

 20 0 0 0 0 8 0 0 0 0 9 0 0 0 0 6
,
 0 1 0 0 52 0 0 0 0 0 0 1 0 0 52 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
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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