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G = C8×D29order 464 = 24·29

Direct product of C8 and D29

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C8×D29, C2323C2, D58.4C4, C4.12D58, Dic29.4C4, C116.12C22, C293(C2×C8), C292C86C2, C58.8(C2×C4), C2.1(C4×D29), (C4×D29).7C2, SmallGroup(464,4)

Series: Derived Chief Lower central Upper central

C1C29 — C8×D29
C1C29C58C116C4×D29 — C8×D29
C29 — C8×D29
C1C8

Generators and relations for C8×D29
 G = < a,b,c | a8=b29=c2=1, ab=ba, ac=ca, cbc=b-1 >

29C2
29C2
29C22
29C4
29C2×C4
29C8
29C2×C8

Smallest permutation representation of C8×D29
On 232 points
Generators in S232
(1 216 109 158 51 188 73 139)(2 217 110 159 52 189 74 140)(3 218 111 160 53 190 75 141)(4 219 112 161 54 191 76 142)(5 220 113 162 55 192 77 143)(6 221 114 163 56 193 78 144)(7 222 115 164 57 194 79 145)(8 223 116 165 58 195 80 117)(9 224 88 166 30 196 81 118)(10 225 89 167 31 197 82 119)(11 226 90 168 32 198 83 120)(12 227 91 169 33 199 84 121)(13 228 92 170 34 200 85 122)(14 229 93 171 35 201 86 123)(15 230 94 172 36 202 87 124)(16 231 95 173 37 203 59 125)(17 232 96 174 38 175 60 126)(18 204 97 146 39 176 61 127)(19 205 98 147 40 177 62 128)(20 206 99 148 41 178 63 129)(21 207 100 149 42 179 64 130)(22 208 101 150 43 180 65 131)(23 209 102 151 44 181 66 132)(24 210 103 152 45 182 67 133)(25 211 104 153 46 183 68 134)(26 212 105 154 47 184 69 135)(27 213 106 155 48 185 70 136)(28 214 107 156 49 186 71 137)(29 215 108 157 50 187 72 138)
(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)(204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232)
(1 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(117 150)(118 149)(119 148)(120 147)(121 146)(122 174)(123 173)(124 172)(125 171)(126 170)(127 169)(128 168)(129 167)(130 166)(131 165)(132 164)(133 163)(134 162)(135 161)(136 160)(137 159)(138 158)(139 157)(140 156)(141 155)(142 154)(143 153)(144 152)(145 151)(175 228)(176 227)(177 226)(178 225)(179 224)(180 223)(181 222)(182 221)(183 220)(184 219)(185 218)(186 217)(187 216)(188 215)(189 214)(190 213)(191 212)(192 211)(193 210)(194 209)(195 208)(196 207)(197 206)(198 205)(199 204)(200 232)(201 231)(202 230)(203 229)

G:=sub<Sym(232)| (1,216,109,158,51,188,73,139)(2,217,110,159,52,189,74,140)(3,218,111,160,53,190,75,141)(4,219,112,161,54,191,76,142)(5,220,113,162,55,192,77,143)(6,221,114,163,56,193,78,144)(7,222,115,164,57,194,79,145)(8,223,116,165,58,195,80,117)(9,224,88,166,30,196,81,118)(10,225,89,167,31,197,82,119)(11,226,90,168,32,198,83,120)(12,227,91,169,33,199,84,121)(13,228,92,170,34,200,85,122)(14,229,93,171,35,201,86,123)(15,230,94,172,36,202,87,124)(16,231,95,173,37,203,59,125)(17,232,96,174,38,175,60,126)(18,204,97,146,39,176,61,127)(19,205,98,147,40,177,62,128)(20,206,99,148,41,178,63,129)(21,207,100,149,42,179,64,130)(22,208,101,150,43,180,65,131)(23,209,102,151,44,181,66,132)(24,210,103,152,45,182,67,133)(25,211,104,153,46,183,68,134)(26,212,105,154,47,184,69,135)(27,213,106,155,48,185,70,136)(28,214,107,156,49,186,71,137)(29,215,108,157,50,187,72,138), (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)(204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(117,150)(118,149)(119,148)(120,147)(121,146)(122,174)(123,173)(124,172)(125,171)(126,170)(127,169)(128,168)(129,167)(130,166)(131,165)(132,164)(133,163)(134,162)(135,161)(136,160)(137,159)(138,158)(139,157)(140,156)(141,155)(142,154)(143,153)(144,152)(145,151)(175,228)(176,227)(177,226)(178,225)(179,224)(180,223)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,210)(194,209)(195,208)(196,207)(197,206)(198,205)(199,204)(200,232)(201,231)(202,230)(203,229)>;

G:=Group( (1,216,109,158,51,188,73,139)(2,217,110,159,52,189,74,140)(3,218,111,160,53,190,75,141)(4,219,112,161,54,191,76,142)(5,220,113,162,55,192,77,143)(6,221,114,163,56,193,78,144)(7,222,115,164,57,194,79,145)(8,223,116,165,58,195,80,117)(9,224,88,166,30,196,81,118)(10,225,89,167,31,197,82,119)(11,226,90,168,32,198,83,120)(12,227,91,169,33,199,84,121)(13,228,92,170,34,200,85,122)(14,229,93,171,35,201,86,123)(15,230,94,172,36,202,87,124)(16,231,95,173,37,203,59,125)(17,232,96,174,38,175,60,126)(18,204,97,146,39,176,61,127)(19,205,98,147,40,177,62,128)(20,206,99,148,41,178,63,129)(21,207,100,149,42,179,64,130)(22,208,101,150,43,180,65,131)(23,209,102,151,44,181,66,132)(24,210,103,152,45,182,67,133)(25,211,104,153,46,183,68,134)(26,212,105,154,47,184,69,135)(27,213,106,155,48,185,70,136)(28,214,107,156,49,186,71,137)(29,215,108,157,50,187,72,138), (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)(204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(117,150)(118,149)(119,148)(120,147)(121,146)(122,174)(123,173)(124,172)(125,171)(126,170)(127,169)(128,168)(129,167)(130,166)(131,165)(132,164)(133,163)(134,162)(135,161)(136,160)(137,159)(138,158)(139,157)(140,156)(141,155)(142,154)(143,153)(144,152)(145,151)(175,228)(176,227)(177,226)(178,225)(179,224)(180,223)(181,222)(182,221)(183,220)(184,219)(185,218)(186,217)(187,216)(188,215)(189,214)(190,213)(191,212)(192,211)(193,210)(194,209)(195,208)(196,207)(197,206)(198,205)(199,204)(200,232)(201,231)(202,230)(203,229) );

G=PermutationGroup([(1,216,109,158,51,188,73,139),(2,217,110,159,52,189,74,140),(3,218,111,160,53,190,75,141),(4,219,112,161,54,191,76,142),(5,220,113,162,55,192,77,143),(6,221,114,163,56,193,78,144),(7,222,115,164,57,194,79,145),(8,223,116,165,58,195,80,117),(9,224,88,166,30,196,81,118),(10,225,89,167,31,197,82,119),(11,226,90,168,32,198,83,120),(12,227,91,169,33,199,84,121),(13,228,92,170,34,200,85,122),(14,229,93,171,35,201,86,123),(15,230,94,172,36,202,87,124),(16,231,95,173,37,203,59,125),(17,232,96,174,38,175,60,126),(18,204,97,146,39,176,61,127),(19,205,98,147,40,177,62,128),(20,206,99,148,41,178,63,129),(21,207,100,149,42,179,64,130),(22,208,101,150,43,180,65,131),(23,209,102,151,44,181,66,132),(24,210,103,152,45,182,67,133),(25,211,104,153,46,183,68,134),(26,212,105,154,47,184,69,135),(27,213,106,155,48,185,70,136),(28,214,107,156,49,186,71,137),(29,215,108,157,50,187,72,138)], [(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),(204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,58),(23,57),(24,56),(25,55),(26,54),(27,53),(28,52),(29,51),(59,93),(60,92),(61,91),(62,90),(63,89),(64,88),(65,116),(66,115),(67,114),(68,113),(69,112),(70,111),(71,110),(72,109),(73,108),(74,107),(75,106),(76,105),(77,104),(78,103),(79,102),(80,101),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(117,150),(118,149),(119,148),(120,147),(121,146),(122,174),(123,173),(124,172),(125,171),(126,170),(127,169),(128,168),(129,167),(130,166),(131,165),(132,164),(133,163),(134,162),(135,161),(136,160),(137,159),(138,158),(139,157),(140,156),(141,155),(142,154),(143,153),(144,152),(145,151),(175,228),(176,227),(177,226),(178,225),(179,224),(180,223),(181,222),(182,221),(183,220),(184,219),(185,218),(186,217),(187,216),(188,215),(189,214),(190,213),(191,212),(192,211),(193,210),(194,209),(195,208),(196,207),(197,206),(198,205),(199,204),(200,232),(201,231),(202,230),(203,229)])

128 conjugacy classes

class 1 2A2B2C4A4B4C4D8A8B8C8D8E8F8G8H29A···29N58A···58N116A···116AB232A···232BD
order122244448888888829···2958···58116···116232···232
size1129291129291111292929292···22···22···22···2

128 irreducible representations

dim11111112222
type++++++
imageC1C2C2C2C4C4C8D29D58C4×D29C8×D29
kernelC8×D29C292C8C232C4×D29Dic29D58D29C8C4C2C1
# reps111122814142856

Matrix representation of C8×D29 in GL2(𝔽233) generated by

2210
0221
,
831
2320
,
0232
2320
G:=sub<GL(2,GF(233))| [221,0,0,221],[83,232,1,0],[0,232,232,0] >;

C8×D29 in GAP, Magma, Sage, TeX

C_8\times D_{29}
% in TeX

G:=Group("C8xD29");
// GroupNames label

G:=SmallGroup(464,4);
// by ID

G=gap.SmallGroup(464,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-29,26,42,11204]);
// Polycyclic

G:=Group<a,b,c|a^8=b^29=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8×D29 in TeX

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