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## G = C4○D4×C29order 464 = 24·29

### Direct product of C29 and C4○D4

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

Aliases: C4○D4×C29, D42C58, Q82C58, C58.13C23, C116.21C22, (C2×C4)⋊3C58, (C2×C116)⋊7C2, (D4×C29)⋊5C2, C4.5(C2×C58), (Q8×C29)⋊5C2, C22.(C2×C58), C2.3(C22×C58), (C2×C58).2C22, SmallGroup(464,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C29
 Chief series C1 — C2 — C58 — C2×C58 — D4×C29 — C4○D4×C29
 Lower central C1 — C2 — C4○D4×C29
 Upper central C1 — C116 — C4○D4×C29

Generators and relations for C4○D4×C29
G = < a,b,c,d | a29=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Smallest permutation representation of C4○D4×C29
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 78 108 162)(2 79 109 163)(3 80 110 164)(4 81 111 165)(5 82 112 166)(6 83 113 167)(7 84 114 168)(8 85 115 169)(9 86 116 170)(10 87 88 171)(11 59 89 172)(12 60 90 173)(13 61 91 174)(14 62 92 146)(15 63 93 147)(16 64 94 148)(17 65 95 149)(18 66 96 150)(19 67 97 151)(20 68 98 152)(21 69 99 153)(22 70 100 154)(23 71 101 155)(24 72 102 156)(25 73 103 157)(26 74 104 158)(27 75 105 159)(28 76 106 160)(29 77 107 161)(30 202 206 141)(31 203 207 142)(32 175 208 143)(33 176 209 144)(34 177 210 145)(35 178 211 117)(36 179 212 118)(37 180 213 119)(38 181 214 120)(39 182 215 121)(40 183 216 122)(41 184 217 123)(42 185 218 124)(43 186 219 125)(44 187 220 126)(45 188 221 127)(46 189 222 128)(47 190 223 129)(48 191 224 130)(49 192 225 131)(50 193 226 132)(51 194 227 133)(52 195 228 134)(53 196 229 135)(54 197 230 136)(55 198 231 137)(56 199 232 138)(57 200 204 139)(58 201 205 140)
(1 195 108 134)(2 196 109 135)(3 197 110 136)(4 198 111 137)(5 199 112 138)(6 200 113 139)(7 201 114 140)(8 202 115 141)(9 203 116 142)(10 175 88 143)(11 176 89 144)(12 177 90 145)(13 178 91 117)(14 179 92 118)(15 180 93 119)(16 181 94 120)(17 182 95 121)(18 183 96 122)(19 184 97 123)(20 185 98 124)(21 186 99 125)(22 187 100 126)(23 188 101 127)(24 189 102 128)(25 190 103 129)(26 191 104 130)(27 192 105 131)(28 193 106 132)(29 194 107 133)(30 85 206 169)(31 86 207 170)(32 87 208 171)(33 59 209 172)(34 60 210 173)(35 61 211 174)(36 62 212 146)(37 63 213 147)(38 64 214 148)(39 65 215 149)(40 66 216 150)(41 67 217 151)(42 68 218 152)(43 69 219 153)(44 70 220 154)(45 71 221 155)(46 72 222 156)(47 73 223 157)(48 74 224 158)(49 75 225 159)(50 76 226 160)(51 77 227 161)(52 78 228 162)(53 79 229 163)(54 80 230 164)(55 81 231 165)(56 82 232 166)(57 83 204 167)(58 84 205 168)
(30 206)(31 207)(32 208)(33 209)(34 210)(35 211)(36 212)(37 213)(38 214)(39 215)(40 216)(41 217)(42 218)(43 219)(44 220)(45 221)(46 222)(47 223)(48 224)(49 225)(50 226)(51 227)(52 228)(53 229)(54 230)(55 231)(56 232)(57 204)(58 205)(117 178)(118 179)(119 180)(120 181)(121 182)(122 183)(123 184)(124 185)(125 186)(126 187)(127 188)(128 189)(129 190)(130 191)(131 192)(132 193)(133 194)(134 195)(135 196)(136 197)(137 198)(138 199)(139 200)(140 201)(141 202)(142 203)(143 175)(144 176)(145 177)

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

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

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

290 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 29A ··· 29AB 58A ··· 58AB 58AC ··· 58DH 116A ··· 116BD 116BE ··· 116EJ order 1 2 2 2 2 4 4 4 4 4 29 ··· 29 58 ··· 58 58 ··· 58 116 ··· 116 116 ··· 116 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 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 C4○D4 C4○D4×C29 kernel C4○D4×C29 C2×C116 D4×C29 Q8×C29 C4○D4 C2×C4 D4 Q8 C29 C1 # reps 1 3 3 1 28 84 84 28 2 56

Matrix representation of C4○D4×C29 in GL2(𝔽233) generated by

 2 0 0 2
,
 89 0 0 89
,
 145 28 231 88
,
 1 145 0 232
G:=sub<GL(2,GF(233))| [2,0,0,2],[89,0,0,89],[145,231,28,88],[1,0,145,232] >;

C4○D4×C29 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{29}
% in TeX

G:=Group("C4oD4xC29");
// GroupNames label

G:=SmallGroup(464,48);
// by ID

G=gap.SmallGroup(464,48);
# by ID

G:=PCGroup([5,-2,-2,-2,-29,-2,2341,882]);
// Polycyclic

G:=Group<a,b,c,d|a^29=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

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