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## G = D4×C58order 464 = 24·29

### Direct product of C58 and D4

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

Aliases: D4×C58, C23⋊C58, C1164C22, C58.11C23, C4⋊(C2×C58), (C2×C4)⋊2C58, C22⋊(C2×C58), (C2×C116)⋊6C2, (C22×C58)⋊1C2, (C2×C58)⋊2C22, C2.1(C22×C58), SmallGroup(464,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C58
 Chief series C1 — C2 — C58 — C2×C58 — D4×C29 — D4×C58
 Lower central C1 — C2 — D4×C58
 Upper central C1 — C2×C58 — D4×C58

Generators and relations for D4×C58
G = < a,b,c | a58=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C2×D4, C29, C58, C58, C58, C116, C2×C58, C2×C58, C2×C58, C2×C116, D4×C29, C22×C58, D4×C58
Quotients: C1, C2, C22, D4, C23, C2×D4, C29, C58, C2×C58, D4×C29, C22×C58, D4×C58

Smallest permutation representation of D4×C58
On 232 points
Generators in S232
(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 84 190 174)(2 85 191 117)(3 86 192 118)(4 87 193 119)(5 88 194 120)(6 89 195 121)(7 90 196 122)(8 91 197 123)(9 92 198 124)(10 93 199 125)(11 94 200 126)(12 95 201 127)(13 96 202 128)(14 97 203 129)(15 98 204 130)(16 99 205 131)(17 100 206 132)(18 101 207 133)(19 102 208 134)(20 103 209 135)(21 104 210 136)(22 105 211 137)(23 106 212 138)(24 107 213 139)(25 108 214 140)(26 109 215 141)(27 110 216 142)(28 111 217 143)(29 112 218 144)(30 113 219 145)(31 114 220 146)(32 115 221 147)(33 116 222 148)(34 59 223 149)(35 60 224 150)(36 61 225 151)(37 62 226 152)(38 63 227 153)(39 64 228 154)(40 65 229 155)(41 66 230 156)(42 67 231 157)(43 68 232 158)(44 69 175 159)(45 70 176 160)(46 71 177 161)(47 72 178 162)(48 73 179 163)(49 74 180 164)(50 75 181 165)(51 76 182 166)(52 77 183 167)(53 78 184 168)(54 79 185 169)(55 80 186 170)(56 81 187 171)(57 82 188 172)(58 83 189 173)
(1 219)(2 220)(3 221)(4 222)(5 223)(6 224)(7 225)(8 226)(9 227)(10 228)(11 229)(12 230)(13 231)(14 232)(15 175)(16 176)(17 177)(18 178)(19 179)(20 180)(21 181)(22 182)(23 183)(24 184)(25 185)(26 186)(27 187)(28 188)(29 189)(30 190)(31 191)(32 192)(33 193)(34 194)(35 195)(36 196)(37 197)(38 198)(39 199)(40 200)(41 201)(42 202)(43 203)(44 204)(45 205)(46 206)(47 207)(48 208)(49 209)(50 210)(51 211)(52 212)(53 213)(54 214)(55 215)(56 216)(57 217)(58 218)(59 88)(60 89)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(71 100)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)(81 110)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(117 146)(118 147)(119 148)(120 149)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 161)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 169)(141 170)(142 171)(143 172)(144 173)(145 174)

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

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

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

290 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 29A ··· 29AB 58A ··· 58CF 58CG ··· 58GN 116A ··· 116BD order 1 2 2 2 2 2 2 2 4 4 29 ··· 29 58 ··· 58 58 ··· 58 116 ··· 116 size 1 1 1 1 2 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

290 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C29 C58 C58 C58 D4 D4×C29 kernel D4×C58 C2×C116 D4×C29 C22×C58 C2×D4 C2×C4 D4 C23 C58 C2 # reps 1 1 4 2 28 28 112 56 2 56

Matrix representation of D4×C58 in GL3(𝔽233) generated by

 232 0 0 0 135 0 0 0 135
,
 232 0 0 0 0 232 0 1 0
,
 1 0 0 0 232 0 0 0 1
G:=sub<GL(3,GF(233))| [232,0,0,0,135,0,0,0,135],[232,0,0,0,0,1,0,232,0],[1,0,0,0,232,0,0,0,1] >;

D4×C58 in GAP, Magma, Sage, TeX

D_4\times C_{58}
% in TeX

G:=Group("D4xC58");
// GroupNames label

G:=SmallGroup(464,46);
// by ID

G=gap.SmallGroup(464,46);
# by ID

G:=PCGroup([5,-2,-2,-2,-29,-2,2341]);
// Polycyclic

G:=Group<a,b,c|a^58=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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