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## G = D4×C52order 416 = 25·13

### Direct product of C52 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C52
 Chief series C1 — C2 — C22 — C2×C26 — C2×C52 — C13×C22⋊C4 — D4×C52
 Lower central C1 — C2 — D4×C52
 Upper central C1 — C2×C52 — D4×C52

Generators and relations for D4×C52
G = < a,b,c | a52=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 124 in 94 conjugacy classes, 64 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×3], C22, C22 [×4], C22 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×4], C23 [×2], C13, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C26 [×3], C26 [×4], C4×D4, C52 [×4], C52 [×3], C2×C26, C2×C26 [×4], C2×C26 [×4], C2×C52 [×3], C2×C52 [×2], C2×C52 [×4], D4×C13 [×4], C22×C26 [×2], C4×C52, C13×C22⋊C4 [×2], C13×C4⋊C4, C22×C52 [×2], D4×C26, D4×C52
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C13, C22×C4, C2×D4, C4○D4, C26 [×7], C4×D4, C52 [×4], C2×C26 [×7], C2×C52 [×6], D4×C13 [×2], C22×C26, C22×C52, D4×C26, C13×C4○D4, D4×C52

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

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

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

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

260 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4L 13A ··· 13L 26A ··· 26AJ 26AK ··· 26CF 52A ··· 52AV 52AW ··· 52EN order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 size 1 1 1 1 2 2 2 2 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

260 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C13 C26 C26 C26 C26 C26 C52 D4 C4○D4 D4×C13 C13×C4○D4 kernel D4×C52 C4×C52 C13×C22⋊C4 C13×C4⋊C4 C22×C52 D4×C26 D4×C13 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C52 C26 C4 C2 # reps 1 1 2 1 2 1 8 12 12 24 12 24 12 96 2 2 24 24

Matrix representation of D4×C52 in GL3(𝔽53) generated by

 30 0 0 0 27 0 0 0 27
,
 1 0 0 0 0 1 0 52 0
,
 52 0 0 0 1 0 0 0 52
G:=sub<GL(3,GF(53))| [30,0,0,0,27,0,0,0,27],[1,0,0,0,0,52,0,1,0],[52,0,0,0,1,0,0,0,52] >;

D4×C52 in GAP, Magma, Sage, TeX

D_4\times C_{52}
% in TeX

G:=Group("D4xC52");
// GroupNames label

G:=SmallGroup(416,179);
// by ID

G=gap.SmallGroup(416,179);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1248,1273,950]);
// Polycyclic

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

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