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G = C61⋊D4order 488 = 23·61

The semidirect product of C61 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C612D4, C22⋊D61, Dic61⋊C2, D1222C2, C2.5D122, C122.5C22, (C2×C122)⋊2C2, SmallGroup(488,8)

Series: Derived Chief Lower central Upper central

C1C122 — C61⋊D4
C1C61C122D122 — C61⋊D4
C61C122 — C61⋊D4
C1C2C22

Generators and relations for C61⋊D4
 G = < a,b,c | a61=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
122C2
61C4
61C22
2D61
2C122
61D4

Smallest permutation representation of C61⋊D4
On 244 points
Generators in S244
(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)(184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244)
(1 152 87 231)(2 151 88 230)(3 150 89 229)(4 149 90 228)(5 148 91 227)(6 147 92 226)(7 146 93 225)(8 145 94 224)(9 144 95 223)(10 143 96 222)(11 142 97 221)(12 141 98 220)(13 140 99 219)(14 139 100 218)(15 138 101 217)(16 137 102 216)(17 136 103 215)(18 135 104 214)(19 134 105 213)(20 133 106 212)(21 132 107 211)(22 131 108 210)(23 130 109 209)(24 129 110 208)(25 128 111 207)(26 127 112 206)(27 126 113 205)(28 125 114 204)(29 124 115 203)(30 123 116 202)(31 183 117 201)(32 182 118 200)(33 181 119 199)(34 180 120 198)(35 179 121 197)(36 178 122 196)(37 177 62 195)(38 176 63 194)(39 175 64 193)(40 174 65 192)(41 173 66 191)(42 172 67 190)(43 171 68 189)(44 170 69 188)(45 169 70 187)(46 168 71 186)(47 167 72 185)(48 166 73 184)(49 165 74 244)(50 164 75 243)(51 163 76 242)(52 162 77 241)(53 161 78 240)(54 160 79 239)(55 159 80 238)(56 158 81 237)(57 157 82 236)(58 156 83 235)(59 155 84 234)(60 154 85 233)(61 153 86 232)
(2 61)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(62 112)(63 111)(64 110)(65 109)(66 108)(67 107)(68 106)(69 105)(70 104)(71 103)(72 102)(73 101)(74 100)(75 99)(76 98)(77 97)(78 96)(79 95)(80 94)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(113 122)(114 121)(115 120)(116 119)(117 118)(123 199)(124 198)(125 197)(126 196)(127 195)(128 194)(129 193)(130 192)(131 191)(132 190)(133 189)(134 188)(135 187)(136 186)(137 185)(138 184)(139 244)(140 243)(141 242)(142 241)(143 240)(144 239)(145 238)(146 237)(147 236)(148 235)(149 234)(150 233)(151 232)(152 231)(153 230)(154 229)(155 228)(156 227)(157 226)(158 225)(159 224)(160 223)(161 222)(162 221)(163 220)(164 219)(165 218)(166 217)(167 216)(168 215)(169 214)(170 213)(171 212)(172 211)(173 210)(174 209)(175 208)(176 207)(177 206)(178 205)(179 204)(180 203)(181 202)(182 201)(183 200)

G:=sub<Sym(244)| (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)(184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244), (1,152,87,231)(2,151,88,230)(3,150,89,229)(4,149,90,228)(5,148,91,227)(6,147,92,226)(7,146,93,225)(8,145,94,224)(9,144,95,223)(10,143,96,222)(11,142,97,221)(12,141,98,220)(13,140,99,219)(14,139,100,218)(15,138,101,217)(16,137,102,216)(17,136,103,215)(18,135,104,214)(19,134,105,213)(20,133,106,212)(21,132,107,211)(22,131,108,210)(23,130,109,209)(24,129,110,208)(25,128,111,207)(26,127,112,206)(27,126,113,205)(28,125,114,204)(29,124,115,203)(30,123,116,202)(31,183,117,201)(32,182,118,200)(33,181,119,199)(34,180,120,198)(35,179,121,197)(36,178,122,196)(37,177,62,195)(38,176,63,194)(39,175,64,193)(40,174,65,192)(41,173,66,191)(42,172,67,190)(43,171,68,189)(44,170,69,188)(45,169,70,187)(46,168,71,186)(47,167,72,185)(48,166,73,184)(49,165,74,244)(50,164,75,243)(51,163,76,242)(52,162,77,241)(53,161,78,240)(54,160,79,239)(55,159,80,238)(56,158,81,237)(57,157,82,236)(58,156,83,235)(59,155,84,234)(60,154,85,233)(61,153,86,232), (2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(62,112)(63,111)(64,110)(65,109)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(113,122)(114,121)(115,120)(116,119)(117,118)(123,199)(124,198)(125,197)(126,196)(127,195)(128,194)(129,193)(130,192)(131,191)(132,190)(133,189)(134,188)(135,187)(136,186)(137,185)(138,184)(139,244)(140,243)(141,242)(142,241)(143,240)(144,239)(145,238)(146,237)(147,236)(148,235)(149,234)(150,233)(151,232)(152,231)(153,230)(154,229)(155,228)(156,227)(157,226)(158,225)(159,224)(160,223)(161,222)(162,221)(163,220)(164,219)(165,218)(166,217)(167,216)(168,215)(169,214)(170,213)(171,212)(172,211)(173,210)(174,209)(175,208)(176,207)(177,206)(178,205)(179,204)(180,203)(181,202)(182,201)(183,200)>;

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)(184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244), (1,152,87,231)(2,151,88,230)(3,150,89,229)(4,149,90,228)(5,148,91,227)(6,147,92,226)(7,146,93,225)(8,145,94,224)(9,144,95,223)(10,143,96,222)(11,142,97,221)(12,141,98,220)(13,140,99,219)(14,139,100,218)(15,138,101,217)(16,137,102,216)(17,136,103,215)(18,135,104,214)(19,134,105,213)(20,133,106,212)(21,132,107,211)(22,131,108,210)(23,130,109,209)(24,129,110,208)(25,128,111,207)(26,127,112,206)(27,126,113,205)(28,125,114,204)(29,124,115,203)(30,123,116,202)(31,183,117,201)(32,182,118,200)(33,181,119,199)(34,180,120,198)(35,179,121,197)(36,178,122,196)(37,177,62,195)(38,176,63,194)(39,175,64,193)(40,174,65,192)(41,173,66,191)(42,172,67,190)(43,171,68,189)(44,170,69,188)(45,169,70,187)(46,168,71,186)(47,167,72,185)(48,166,73,184)(49,165,74,244)(50,164,75,243)(51,163,76,242)(52,162,77,241)(53,161,78,240)(54,160,79,239)(55,159,80,238)(56,158,81,237)(57,157,82,236)(58,156,83,235)(59,155,84,234)(60,154,85,233)(61,153,86,232), (2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(62,112)(63,111)(64,110)(65,109)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(113,122)(114,121)(115,120)(116,119)(117,118)(123,199)(124,198)(125,197)(126,196)(127,195)(128,194)(129,193)(130,192)(131,191)(132,190)(133,189)(134,188)(135,187)(136,186)(137,185)(138,184)(139,244)(140,243)(141,242)(142,241)(143,240)(144,239)(145,238)(146,237)(147,236)(148,235)(149,234)(150,233)(151,232)(152,231)(153,230)(154,229)(155,228)(156,227)(157,226)(158,225)(159,224)(160,223)(161,222)(162,221)(163,220)(164,219)(165,218)(166,217)(167,216)(168,215)(169,214)(170,213)(171,212)(172,211)(173,210)(174,209)(175,208)(176,207)(177,206)(178,205)(179,204)(180,203)(181,202)(182,201)(183,200) );

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),(184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244)], [(1,152,87,231),(2,151,88,230),(3,150,89,229),(4,149,90,228),(5,148,91,227),(6,147,92,226),(7,146,93,225),(8,145,94,224),(9,144,95,223),(10,143,96,222),(11,142,97,221),(12,141,98,220),(13,140,99,219),(14,139,100,218),(15,138,101,217),(16,137,102,216),(17,136,103,215),(18,135,104,214),(19,134,105,213),(20,133,106,212),(21,132,107,211),(22,131,108,210),(23,130,109,209),(24,129,110,208),(25,128,111,207),(26,127,112,206),(27,126,113,205),(28,125,114,204),(29,124,115,203),(30,123,116,202),(31,183,117,201),(32,182,118,200),(33,181,119,199),(34,180,120,198),(35,179,121,197),(36,178,122,196),(37,177,62,195),(38,176,63,194),(39,175,64,193),(40,174,65,192),(41,173,66,191),(42,172,67,190),(43,171,68,189),(44,170,69,188),(45,169,70,187),(46,168,71,186),(47,167,72,185),(48,166,73,184),(49,165,74,244),(50,164,75,243),(51,163,76,242),(52,162,77,241),(53,161,78,240),(54,160,79,239),(55,159,80,238),(56,158,81,237),(57,157,82,236),(58,156,83,235),(59,155,84,234),(60,154,85,233),(61,153,86,232)], [(2,61),(3,60),(4,59),(5,58),(6,57),(7,56),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(31,32),(62,112),(63,111),(64,110),(65,109),(66,108),(67,107),(68,106),(69,105),(70,104),(71,103),(72,102),(73,101),(74,100),(75,99),(76,98),(77,97),(78,96),(79,95),(80,94),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(113,122),(114,121),(115,120),(116,119),(117,118),(123,199),(124,198),(125,197),(126,196),(127,195),(128,194),(129,193),(130,192),(131,191),(132,190),(133,189),(134,188),(135,187),(136,186),(137,185),(138,184),(139,244),(140,243),(141,242),(142,241),(143,240),(144,239),(145,238),(146,237),(147,236),(148,235),(149,234),(150,233),(151,232),(152,231),(153,230),(154,229),(155,228),(156,227),(157,226),(158,225),(159,224),(160,223),(161,222),(162,221),(163,220),(164,219),(165,218),(166,217),(167,216),(168,215),(169,214),(170,213),(171,212),(172,211),(173,210),(174,209),(175,208),(176,207),(177,206),(178,205),(179,204),(180,203),(181,202),(182,201),(183,200)])

125 conjugacy classes

class 1 2A2B2C 4 61A···61AD122A···122CL
order1222461···61122···122
size1121221222···22···2

125 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D61D122C61⋊D4
kernelC61⋊D4Dic61D122C2×C122C61C22C2C1
# reps11111303060

Matrix representation of C61⋊D4 in GL2(𝔽733) generated by

01
732709
,
197444
114536
,
10
709732
G:=sub<GL(2,GF(733))| [0,732,1,709],[197,114,444,536],[1,709,0,732] >;

C61⋊D4 in GAP, Magma, Sage, TeX

C_{61}\rtimes D_4
% in TeX

G:=Group("C61:D4");
// GroupNames label

G:=SmallGroup(488,8);
// by ID

G=gap.SmallGroup(488,8);
# by ID

G:=PCGroup([4,-2,-2,-2,-61,49,7683]);
// Polycyclic

G:=Group<a,b,c|a^61=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C61⋊D4 in TeX

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