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G = C29⋊C8order 232 = 23·29

The semidirect product of C29 and C8 acting via C8/C2=C4

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C29⋊C8, C58.C4, Dic29.2C2, C2.(C29⋊C4), SmallGroup(232,3)

Series: Derived Chief Lower central Upper central

C1C29 — C29⋊C8
C1C29C58Dic29 — C29⋊C8
C29 — C29⋊C8
C1C2

Generators and relations for C29⋊C8
 G = < a,b | a29=b8=1, bab-1=a17 >

29C4
29C8

Character table of C29⋊C8

 class 124A4B8A8B8C8D29A29B29C29D29E29F29G58A58B58C58D58E58F58G
 size 1129292929292944444444444444
ρ11111111111111111111111    trivial
ρ21111-1-1-1-111111111111111    linear of order 2
ρ311-1-1i-ii-i11111111111111    linear of order 4
ρ411-1-1-ii-ii11111111111111    linear of order 4
ρ51-1-iiζ8ζ83ζ85ζ871111111-1-1-1-1-1-1-1    linear of order 8
ρ61-1i-iζ83ζ8ζ87ζ851111111-1-1-1-1-1-1-1    linear of order 8
ρ71-1i-iζ87ζ85ζ83ζ81111111-1-1-1-1-1-1-1    linear of order 8
ρ81-1-iiζ85ζ87ζ8ζ831111111-1-1-1-1-1-1-1    linear of order 8
ρ944000000ζ2918291629132911ζ29262922297293ζ292529192910294ζ292329152914296ζ29212920299298ζ29282917291229ζ29272924295292ζ29272924295292ζ2918291629132911ζ29262922297293ζ292529192910294ζ292329152914296ζ29212920299298ζ29282917291229    orthogonal lifted from C29⋊C4
ρ1044000000ζ292529192910294ζ29212920299298ζ29282917291229ζ2918291629132911ζ29272924295292ζ29262922297293ζ292329152914296ζ292329152914296ζ292529192910294ζ29212920299298ζ29282917291229ζ2918291629132911ζ29272924295292ζ29262922297293    orthogonal lifted from C29⋊C4
ρ1144000000ζ292329152914296ζ29282917291229ζ2918291629132911ζ29272924295292ζ29262922297293ζ292529192910294ζ29212920299298ζ29212920299298ζ292329152914296ζ29282917291229ζ2918291629132911ζ29272924295292ζ29262922297293ζ292529192910294    orthogonal lifted from C29⋊C4
ρ1244000000ζ29262922297293ζ292329152914296ζ29212920299298ζ29282917291229ζ2918291629132911ζ29272924295292ζ292529192910294ζ292529192910294ζ29262922297293ζ292329152914296ζ29212920299298ζ29282917291229ζ2918291629132911ζ29272924295292    orthogonal lifted from C29⋊C4
ρ1344000000ζ29282917291229ζ29272924295292ζ29262922297293ζ292529192910294ζ292329152914296ζ29212920299298ζ2918291629132911ζ2918291629132911ζ29282917291229ζ29272924295292ζ29262922297293ζ292529192910294ζ292329152914296ζ29212920299298    orthogonal lifted from C29⋊C4
ρ1444000000ζ29272924295292ζ292529192910294ζ292329152914296ζ29212920299298ζ29282917291229ζ2918291629132911ζ29262922297293ζ29262922297293ζ29272924295292ζ292529192910294ζ292329152914296ζ29212920299298ζ29282917291229ζ2918291629132911    orthogonal lifted from C29⋊C4
ρ1544000000ζ29212920299298ζ2918291629132911ζ29272924295292ζ29262922297293ζ292529192910294ζ292329152914296ζ29282917291229ζ29282917291229ζ29212920299298ζ2918291629132911ζ29272924295292ζ29262922297293ζ292529192910294ζ292329152914296    orthogonal lifted from C29⋊C4
ρ164-4000000ζ29282917291229ζ29272924295292ζ29262922297293ζ292529192910294ζ292329152914296ζ29212920299298ζ2918291629132911291829162913291129282917291229292729242952922926292229729329252919291029429232915291429629212920299298    symplectic faithful, Schur index 2
ρ174-4000000ζ2918291629132911ζ29262922297293ζ292529192910294ζ292329152914296ζ29212920299298ζ29282917291229ζ29272924295292292729242952922918291629132911292629222972932925291929102942923291529142962921292029929829282917291229    symplectic faithful, Schur index 2
ρ184-4000000ζ292529192910294ζ29212920299298ζ29282917291229ζ2918291629132911ζ29272924295292ζ29262922297293ζ292329152914296292329152914296292529192910294292129202992982928291729122929182916291329112927292429529229262922297293    symplectic faithful, Schur index 2
ρ194-4000000ζ29212920299298ζ2918291629132911ζ29272924295292ζ29262922297293ζ292529192910294ζ292329152914296ζ29282917291229292829172912292921292029929829182916291329112927292429529229262922297293292529192910294292329152914296    symplectic faithful, Schur index 2
ρ204-4000000ζ29272924295292ζ292529192910294ζ292329152914296ζ29212920299298ζ29282917291229ζ2918291629132911ζ29262922297293292629222972932927292429529229252919291029429232915291429629212920299298292829172912292918291629132911    symplectic faithful, Schur index 2
ρ214-4000000ζ292329152914296ζ29282917291229ζ2918291629132911ζ29272924295292ζ29262922297293ζ292529192910294ζ29212920299298292129202992982923291529142962928291729122929182916291329112927292429529229262922297293292529192910294    symplectic faithful, Schur index 2
ρ224-4000000ζ29262922297293ζ292329152914296ζ29212920299298ζ29282917291229ζ2918291629132911ζ29272924295292ζ292529192910294292529192910294292629222972932923291529142962921292029929829282917291229291829162913291129272924295292    symplectic faithful, Schur index 2

Smallest permutation representation of C29⋊C8
Regular action 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)
(1 232 88 174 44 189 59 132)(2 215 116 162 45 201 87 120)(3 227 115 150 46 184 86 137)(4 210 114 167 47 196 85 125)(5 222 113 155 48 179 84 142)(6 205 112 172 49 191 83 130)(7 217 111 160 50 203 82 118)(8 229 110 148 51 186 81 135)(9 212 109 165 52 198 80 123)(10 224 108 153 53 181 79 140)(11 207 107 170 54 193 78 128)(12 219 106 158 55 176 77 145)(13 231 105 146 56 188 76 133)(14 214 104 163 57 200 75 121)(15 226 103 151 58 183 74 138)(16 209 102 168 30 195 73 126)(17 221 101 156 31 178 72 143)(18 204 100 173 32 190 71 131)(19 216 99 161 33 202 70 119)(20 228 98 149 34 185 69 136)(21 211 97 166 35 197 68 124)(22 223 96 154 36 180 67 141)(23 206 95 171 37 192 66 129)(24 218 94 159 38 175 65 117)(25 230 93 147 39 187 64 134)(26 213 92 164 40 199 63 122)(27 225 91 152 41 182 62 139)(28 208 90 169 42 194 61 127)(29 220 89 157 43 177 60 144)

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

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

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

C29⋊C8 is a maximal subgroup of   D29⋊C8  C116.C4  C29⋊M4(2)
C29⋊C8 is a maximal quotient of   C29⋊C16

Matrix representation of C29⋊C8 in GL5(𝔽233)

10000
0232100
0232010
0232001
08421914148
,
1360000
019614916981
0110205102
016322819124
053204146

G:=sub<GL(5,GF(233))| [1,0,0,0,0,0,232,232,232,84,0,1,0,0,219,0,0,1,0,14,0,0,0,1,148],[136,0,0,0,0,0,196,110,163,53,0,149,205,228,204,0,169,10,19,1,0,81,2,124,46] >;

C29⋊C8 in GAP, Magma, Sage, TeX

C_{29}\rtimes C_8
% in TeX

G:=Group("C29:C8");
// GroupNames label

G:=SmallGroup(232,3);
// by ID

G=gap.SmallGroup(232,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-29,8,21,1539,1799]);
// Polycyclic

G:=Group<a,b|a^29=b^8=1,b*a*b^-1=a^17>;
// generators/relations

Export

Subgroup lattice of C29⋊C8 in TeX
Character table of C29⋊C8 in TeX

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