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G = Q8×C29order 232 = 23·29

Direct product of C29 and Q8

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

Aliases: Q8×C29, C4.C58, C116.3C2, C58.7C22, C2.2(C2×C58), SmallGroup(232,11)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C29
C1C2C58C116 — Q8×C29
C1C2 — Q8×C29
C1C58 — Q8×C29

Generators and relations for Q8×C29
 G = < a,b,c | a29=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C29
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)
(1 158 52 90)(2 159 53 91)(3 160 54 92)(4 161 55 93)(5 162 56 94)(6 163 57 95)(7 164 58 96)(8 165 30 97)(9 166 31 98)(10 167 32 99)(11 168 33 100)(12 169 34 101)(13 170 35 102)(14 171 36 103)(15 172 37 104)(16 173 38 105)(17 174 39 106)(18 146 40 107)(19 147 41 108)(20 148 42 109)(21 149 43 110)(22 150 44 111)(23 151 45 112)(24 152 46 113)(25 153 47 114)(26 154 48 115)(27 155 49 116)(28 156 50 88)(29 157 51 89)(59 216 117 192)(60 217 118 193)(61 218 119 194)(62 219 120 195)(63 220 121 196)(64 221 122 197)(65 222 123 198)(66 223 124 199)(67 224 125 200)(68 225 126 201)(69 226 127 202)(70 227 128 203)(71 228 129 175)(72 229 130 176)(73 230 131 177)(74 231 132 178)(75 232 133 179)(76 204 134 180)(77 205 135 181)(78 206 136 182)(79 207 137 183)(80 208 138 184)(81 209 139 185)(82 210 140 186)(83 211 141 187)(84 212 142 188)(85 213 143 189)(86 214 144 190)(87 215 145 191)
(1 142 52 84)(2 143 53 85)(3 144 54 86)(4 145 55 87)(5 117 56 59)(6 118 57 60)(7 119 58 61)(8 120 30 62)(9 121 31 63)(10 122 32 64)(11 123 33 65)(12 124 34 66)(13 125 35 67)(14 126 36 68)(15 127 37 69)(16 128 38 70)(17 129 39 71)(18 130 40 72)(19 131 41 73)(20 132 42 74)(21 133 43 75)(22 134 44 76)(23 135 45 77)(24 136 46 78)(25 137 47 79)(26 138 48 80)(27 139 49 81)(28 140 50 82)(29 141 51 83)(88 186 156 210)(89 187 157 211)(90 188 158 212)(91 189 159 213)(92 190 160 214)(93 191 161 215)(94 192 162 216)(95 193 163 217)(96 194 164 218)(97 195 165 219)(98 196 166 220)(99 197 167 221)(100 198 168 222)(101 199 169 223)(102 200 170 224)(103 201 171 225)(104 202 172 226)(105 203 173 227)(106 175 174 228)(107 176 146 229)(108 177 147 230)(109 178 148 231)(110 179 149 232)(111 180 150 204)(112 181 151 205)(113 182 152 206)(114 183 153 207)(115 184 154 208)(116 185 155 209)

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

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

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

Q8×C29 is a maximal subgroup of   Q8⋊D29  C29⋊Q16  Q82D29

145 conjugacy classes

class 1  2 4A4B4C29A···29AB58A···58AB116A···116CF
order1244429···2958···58116···116
size112221···11···12···2

145 irreducible representations

dim111122
type++-
imageC1C2C29C58Q8Q8×C29
kernelQ8×C29C116Q8C4C29C1
# reps132884128

Matrix representation of Q8×C29 in GL2(𝔽233) generated by

640
064
,
1231
1232
,
13153
206220
G:=sub<GL(2,GF(233))| [64,0,0,64],[1,1,231,232],[13,206,153,220] >;

Q8×C29 in GAP, Magma, Sage, TeX

Q_8\times C_{29}
% in TeX

G:=Group("Q8xC29");
// GroupNames label

G:=SmallGroup(232,11);
// by ID

G=gap.SmallGroup(232,11);
# by ID

G:=PCGroup([4,-2,-2,-29,-2,464,945,469]);
// Polycyclic

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

Export

Subgroup lattice of Q8×C29 in TeX

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