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G = C4.Dic29order 464 = 24·29

The non-split extension by C4 of Dic29 acting via Dic29/C58=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4.Dic29, C116.4C4, C4.15D58, C294M4(2), C22.Dic29, C116.15C22, C292C85C2, (C2×C58).5C4, (C2×C4).2D29, C58.14(C2×C4), (C2×C116).5C2, C2.3(C2×Dic29), SmallGroup(464,10)

Series: Derived Chief Lower central Upper central

C1C58 — C4.Dic29
C1C29C58C116C292C8 — C4.Dic29
C29C58 — C4.Dic29
C1C4C2×C4

Generators and relations for C4.Dic29
 G = < a,b,c | a4=1, b58=a2, c2=b29, ab=ba, cac-1=a-1, cbc-1=b57 >

2C2
2C58
29C8
29C8
29M4(2)

Smallest permutation representation of C4.Dic29
On 232 points
Generators in S232
(1 30 59 88)(2 31 60 89)(3 32 61 90)(4 33 62 91)(5 34 63 92)(6 35 64 93)(7 36 65 94)(8 37 66 95)(9 38 67 96)(10 39 68 97)(11 40 69 98)(12 41 70 99)(13 42 71 100)(14 43 72 101)(15 44 73 102)(16 45 74 103)(17 46 75 104)(18 47 76 105)(19 48 77 106)(20 49 78 107)(21 50 79 108)(22 51 80 109)(23 52 81 110)(24 53 82 111)(25 54 83 112)(26 55 84 113)(27 56 85 114)(28 57 86 115)(29 58 87 116)(117 204 175 146)(118 205 176 147)(119 206 177 148)(120 207 178 149)(121 208 179 150)(122 209 180 151)(123 210 181 152)(124 211 182 153)(125 212 183 154)(126 213 184 155)(127 214 185 156)(128 215 186 157)(129 216 187 158)(130 217 188 159)(131 218 189 160)(132 219 190 161)(133 220 191 162)(134 221 192 163)(135 222 193 164)(136 223 194 165)(137 224 195 166)(138 225 196 167)(139 226 197 168)(140 227 198 169)(141 228 199 170)(142 229 200 171)(143 230 201 172)(144 231 202 173)(145 232 203 174)
(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)
(1 146 30 175 59 204 88 117)(2 203 31 232 60 145 89 174)(3 144 32 173 61 202 90 231)(4 201 33 230 62 143 91 172)(5 142 34 171 63 200 92 229)(6 199 35 228 64 141 93 170)(7 140 36 169 65 198 94 227)(8 197 37 226 66 139 95 168)(9 138 38 167 67 196 96 225)(10 195 39 224 68 137 97 166)(11 136 40 165 69 194 98 223)(12 193 41 222 70 135 99 164)(13 134 42 163 71 192 100 221)(14 191 43 220 72 133 101 162)(15 132 44 161 73 190 102 219)(16 189 45 218 74 131 103 160)(17 130 46 159 75 188 104 217)(18 187 47 216 76 129 105 158)(19 128 48 157 77 186 106 215)(20 185 49 214 78 127 107 156)(21 126 50 155 79 184 108 213)(22 183 51 212 80 125 109 154)(23 124 52 153 81 182 110 211)(24 181 53 210 82 123 111 152)(25 122 54 151 83 180 112 209)(26 179 55 208 84 121 113 150)(27 120 56 149 85 178 114 207)(28 177 57 206 86 119 115 148)(29 118 58 147 87 176 116 205)

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

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

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

122 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D29A···29N58A···58AP116A···116BD
order122444888829···2958···58116···116
size112112585858582···22···22···2

122 irreducible representations

dim11111222222
type++++-+-
imageC1C2C2C4C4M4(2)D29Dic29D58Dic29C4.Dic29
kernelC4.Dic29C292C8C2×C116C116C2×C58C29C2×C4C4C4C22C1
# reps1212221414141456

Matrix representation of C4.Dic29 in GL2(𝔽233) generated by

890
0144
,
2240
026
,
01
890
G:=sub<GL(2,GF(233))| [89,0,0,144],[224,0,0,26],[0,89,1,0] >;

C4.Dic29 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_{29}
% in TeX

G:=Group("C4.Dic29");
// GroupNames label

G:=SmallGroup(464,10);
// by ID

G=gap.SmallGroup(464,10);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-29,20,101,42,11204]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^58=a^2,c^2=b^29,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^57>;
// generators/relations

Export

Subgroup lattice of C4.Dic29 in TeX

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