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## G = D8×C52order 400 = 24·52

### Direct product of C52 and D8

direct product, metacyclic, nilpotent (class 3), monomial

Aliases: D8×C52, C405C10, C4.1C102, D4⋊(C5×C10), (C5×C40)⋊9C2, C81(C5×C10), (C5×D4)⋊4C10, (C5×C10).41D4, C10.20(C5×D4), (D4×C52)⋊7C2, C2.3(D4×C52), C20.23(C2×C10), (C5×C20).50C22, SmallGroup(400,113)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C52
 Chief series C1 — C2 — C4 — C20 — C5×C20 — D4×C52 — D8×C52
 Lower central C1 — C2 — C4 — D8×C52
 Upper central C1 — C5×C10 — C5×C20 — D8×C52

Generators and relations for D8×C52
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 152 in 88 conjugacy classes, 56 normal (10 characteristic)
C1, C2, C2, C4, C22, C5, C8, D4, C10, C10, D8, C20, C2×C10, C52, C40, C5×D4, C5×C10, C5×C10, C5×D8, C5×C20, C102, C5×C40, D4×C52, D8×C52
Quotients: C1, C2, C22, C5, D4, C10, D8, C2×C10, C52, C5×D4, C5×C10, C5×D8, C102, D4×C52, D8×C52

Smallest permutation representation of D8×C52
On 200 points
Generators in S200
(1 103 90 175 66)(2 104 91 176 67)(3 97 92 169 68)(4 98 93 170 69)(5 99 94 171 70)(6 100 95 172 71)(7 101 96 173 72)(8 102 89 174 65)(9 195 179 78 121)(10 196 180 79 122)(11 197 181 80 123)(12 198 182 73 124)(13 199 183 74 125)(14 200 184 75 126)(15 193 177 76 127)(16 194 178 77 128)(17 105 187 41 133)(18 106 188 42 134)(19 107 189 43 135)(20 108 190 44 136)(21 109 191 45 129)(22 110 192 46 130)(23 111 185 47 131)(24 112 186 48 132)(25 117 155 49 137)(26 118 156 50 138)(27 119 157 51 139)(28 120 158 52 140)(29 113 159 53 141)(30 114 160 54 142)(31 115 153 55 143)(32 116 154 56 144)(33 86 163 62 149)(34 87 164 63 150)(35 88 165 64 151)(36 81 166 57 152)(37 82 167 58 145)(38 83 168 59 146)(39 84 161 60 147)(40 85 162 61 148)
(1 159 13 58 109)(2 160 14 59 110)(3 153 15 60 111)(4 154 16 61 112)(5 155 9 62 105)(6 156 10 63 106)(7 157 11 64 107)(8 158 12 57 108)(17 70 117 121 163)(18 71 118 122 164)(19 72 119 123 165)(20 65 120 124 166)(21 66 113 125 167)(22 67 114 126 168)(23 68 115 127 161)(24 69 116 128 162)(25 78 86 133 171)(26 79 87 134 172)(27 80 88 135 173)(28 73 81 136 174)(29 74 82 129 175)(30 75 83 130 176)(31 76 84 131 169)(32 77 85 132 170)(33 41 94 137 179)(34 42 95 138 180)(35 43 96 139 181)(36 44 89 140 182)(37 45 90 141 183)(38 46 91 142 184)(39 47 92 143 177)(40 48 93 144 178)(49 195 149 187 99)(50 196 150 188 100)(51 197 151 189 101)(52 198 152 190 102)(53 199 145 191 103)(54 200 146 192 104)(55 193 147 185 97)(56 194 148 186 98)
(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)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 24)(18 23)(19 22)(20 21)(25 32)(26 31)(27 30)(28 29)(33 40)(34 39)(35 38)(36 37)(41 48)(42 47)(43 46)(44 45)(49 56)(50 55)(51 54)(52 53)(57 58)(59 64)(60 63)(61 62)(65 66)(67 72)(68 71)(69 70)(73 74)(75 80)(76 79)(77 78)(81 82)(83 88)(84 87)(85 86)(89 90)(91 96)(92 95)(93 94)(97 100)(98 99)(101 104)(102 103)(105 112)(106 111)(107 110)(108 109)(113 120)(114 119)(115 118)(116 117)(121 128)(122 127)(123 126)(124 125)(129 136)(130 135)(131 134)(132 133)(137 144)(138 143)(139 142)(140 141)(145 152)(146 151)(147 150)(148 149)(153 156)(154 155)(157 160)(158 159)(161 164)(162 163)(165 168)(166 167)(169 172)(170 171)(173 176)(174 175)(177 180)(178 179)(181 184)(182 183)(185 188)(186 187)(189 192)(190 191)(193 196)(194 195)(197 200)(198 199)

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

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

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

175 conjugacy classes

 class 1 2A 2B 2C 4 5A ··· 5X 8A 8B 10A ··· 10X 10Y ··· 10BT 20A ··· 20X 40A ··· 40AV order 1 2 2 2 4 5 ··· 5 8 8 10 ··· 10 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 4 4 2 1 ··· 1 2 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2

175 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C5 C10 C10 D4 D8 C5×D4 C5×D8 kernel D8×C52 C5×C40 D4×C52 C5×D8 C40 C5×D4 C5×C10 C52 C10 C5 # reps 1 1 2 24 24 48 1 2 24 48

Matrix representation of D8×C52 in GL3(𝔽41) generated by

 1 0 0 0 37 0 0 0 37
,
 37 0 0 0 10 0 0 0 10
,
 40 0 0 0 0 24 0 29 24
,
 1 0 0 0 0 24 0 12 0
G:=sub<GL(3,GF(41))| [1,0,0,0,37,0,0,0,37],[37,0,0,0,10,0,0,0,10],[40,0,0,0,0,29,0,24,24],[1,0,0,0,0,12,0,24,0] >;

D8×C52 in GAP, Magma, Sage, TeX

D_8\times C_5^2
% in TeX

G:=Group("D8xC5^2");
// GroupNames label

G:=SmallGroup(400,113);
// by ID

G=gap.SmallGroup(400,113);
# by ID

G:=PCGroup([6,-2,-2,-5,-5,-2,-2,1225,9004,4510,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽