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## G = C8×C5⋊D5order 400 = 24·52

### Direct product of C8 and C5⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C8×C5⋊D5
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C4×C5⋊D5 — C8×C5⋊D5
 Lower central C52 — C8×C5⋊D5
 Upper central C1 — C8

Generators and relations for C8×C5⋊D5
G = < a,b,c,d | a8=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 424 in 88 conjugacy classes, 39 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D5, C10, C2×C8, Dic5, C20, D10, C52, C52C8, C40, C4×D5, C5⋊D5, C5×C10, C8×D5, C526C4, C5×C20, C2×C5⋊D5, C527C8, C5×C40, C4×C5⋊D5, C8×C5⋊D5
Quotients: C1, C2, C4, C22, C8, C2×C4, D5, C2×C8, D10, C4×D5, C5⋊D5, C8×D5, C2×C5⋊D5, C4×C5⋊D5, C8×C5⋊D5

Smallest permutation representation of C8×C5⋊D5
On 200 points
Generators in S200
(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 23 168 89 104)(2 24 161 90 97)(3 17 162 91 98)(4 18 163 92 99)(5 19 164 93 100)(6 20 165 94 101)(7 21 166 95 102)(8 22 167 96 103)(9 111 34 152 73)(10 112 35 145 74)(11 105 36 146 75)(12 106 37 147 76)(13 107 38 148 77)(14 108 39 149 78)(15 109 40 150 79)(16 110 33 151 80)(25 130 178 115 175)(26 131 179 116 176)(27 132 180 117 169)(28 133 181 118 170)(29 134 182 119 171)(30 135 183 120 172)(31 136 184 113 173)(32 129 177 114 174)(41 195 52 123 61)(42 196 53 124 62)(43 197 54 125 63)(44 198 55 126 64)(45 199 56 127 57)(46 200 49 128 58)(47 193 50 121 59)(48 194 51 122 60)(65 143 189 155 86)(66 144 190 156 87)(67 137 191 157 88)(68 138 192 158 81)(69 139 185 159 82)(70 140 186 160 83)(71 141 187 153 84)(72 142 188 154 85)
(1 115 63 189 36)(2 116 64 190 37)(3 117 57 191 38)(4 118 58 192 39)(5 119 59 185 40)(6 120 60 186 33)(7 113 61 187 34)(8 114 62 188 35)(9 95 136 52 71)(10 96 129 53 72)(11 89 130 54 65)(12 90 131 55 66)(13 91 132 56 67)(14 92 133 49 68)(15 93 134 50 69)(16 94 135 51 70)(17 169 45 157 148)(18 170 46 158 149)(19 171 47 159 150)(20 172 48 160 151)(21 173 41 153 152)(22 174 42 154 145)(23 175 43 155 146)(24 176 44 156 147)(25 197 86 75 168)(26 198 87 76 161)(27 199 88 77 162)(28 200 81 78 163)(29 193 82 79 164)(30 194 83 80 165)(31 195 84 73 166)(32 196 85 74 167)(97 179 126 144 106)(98 180 127 137 107)(99 181 128 138 108)(100 182 121 139 109)(101 183 122 140 110)(102 184 123 141 111)(103 177 124 142 112)(104 178 125 143 105)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 33)(7 34)(8 35)(9 166)(10 167)(11 168)(12 161)(13 162)(14 163)(15 164)(16 165)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 105)(24 106)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 121)(48 122)(49 200)(50 193)(51 194)(52 195)(53 196)(54 197)(55 198)(56 199)(73 95)(74 96)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 133)(82 134)(83 135)(84 136)(85 129)(86 130)(87 131)(88 132)(97 147)(98 148)(99 149)(100 150)(101 151)(102 152)(103 145)(104 146)(113 187)(114 188)(115 189)(116 190)(117 191)(118 192)(119 185)(120 186)(137 169)(138 170)(139 171)(140 172)(141 173)(142 174)(143 175)(144 176)(153 184)(154 177)(155 178)(156 179)(157 180)(158 181)(159 182)(160 183)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A ··· 5L 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10L 20A ··· 20X 40A ··· 40AV order 1 2 2 2 4 4 4 4 5 ··· 5 8 8 8 8 8 8 8 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 25 25 1 1 25 25 2 ··· 2 1 1 1 1 25 25 25 25 2 ··· 2 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 D5 D10 C4×D5 C8×D5 kernel C8×C5⋊D5 C52⋊7C8 C5×C40 C4×C5⋊D5 C52⋊6C4 C2×C5⋊D5 C5⋊D5 C40 C20 C10 C5 # reps 1 1 1 1 2 2 8 12 12 24 48

Matrix representation of C8×C5⋊D5 in GL5(𝔽41)

 14 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 32 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 1 0 0 0 40 6 0 0 0 0 0 0 1 0 0 0 40 6
,
 1 0 0 0 0 0 40 6 0 0 0 35 35 0 0 0 0 0 6 40 0 0 0 1 0
,
 40 0 0 0 0 0 40 6 0 0 0 0 1 0 0 0 0 0 6 6 0 0 0 1 35

G:=sub<GL(5,GF(41))| [14,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,32],[1,0,0,0,0,0,0,40,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,1,6],[1,0,0,0,0,0,40,35,0,0,0,6,35,0,0,0,0,0,6,1,0,0,0,40,0],[40,0,0,0,0,0,40,0,0,0,0,6,1,0,0,0,0,0,6,1,0,0,0,6,35] >;

C8×C5⋊D5 in GAP, Magma, Sage, TeX

C_8\times C_5\rtimes D_5
% in TeX

G:=Group("C8xC5:D5");
// GroupNames label

G:=SmallGroup(400,92);
// by ID

G=gap.SmallGroup(400,92);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,31,50,1924,11525]);
// Polycyclic

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

׿
×
𝔽