metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C55⋊1D4, D22⋊1D5, D10⋊1D11, C10.4D22, C22.4D10, Dic55⋊4C2, C110.4C22, (D5×C22)⋊1C2, C5⋊2(C11⋊D4), C11⋊2(C5⋊D4), C2.4(D5×D11), (C10×D11)⋊1C2, SmallGroup(440,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C55⋊D4
G = < a,b,c | a55=b4=c2=1, bab-1=a-1, cac=a21, cbc=b-1 >
Smallest permutation representation of C55⋊D4
►On 220 points
Generators in S220
(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)
(1 165 104 190)(2 164 105 189)(3 163 106 188)(4 162 107 187)(5 161 108 186)(6 160 109 185)(7 159 110 184)(8 158 56 183)(9 157 57 182)(10 156 58 181)(11 155 59 180)(12 154 60 179)(13 153 61 178)(14 152 62 177)(15 151 63 176)(16 150 64 175)(17 149 65 174)(18 148 66 173)(19 147 67 172)(20 146 68 171)(21 145 69 170)(22 144 70 169)(23 143 71 168)(24 142 72 167)(25 141 73 166)(26 140 74 220)(27 139 75 219)(28 138 76 218)(29 137 77 217)(30 136 78 216)(31 135 79 215)(32 134 80 214)(33 133 81 213)(34 132 82 212)(35 131 83 211)(36 130 84 210)(37 129 85 209)(38 128 86 208)(39 127 87 207)(40 126 88 206)(41 125 89 205)(42 124 90 204)(43 123 91 203)(44 122 92 202)(45 121 93 201)(46 120 94 200)(47 119 95 199)(48 118 96 198)(49 117 97 197)(50 116 98 196)(51 115 99 195)(52 114 100 194)(53 113 101 193)(54 112 102 192)(55 111 103 191)
(2 22)(3 43)(4 9)(5 30)(6 51)(7 17)(8 38)(10 25)(11 46)(13 33)(14 54)(15 20)(16 41)(18 28)(19 49)(21 36)(24 44)(26 31)(27 52)(29 39)(32 47)(35 55)(37 42)(40 50)(48 53)(56 86)(57 107)(58 73)(59 94)(61 81)(62 102)(63 68)(64 89)(65 110)(66 76)(67 97)(69 84)(70 105)(72 92)(74 79)(75 100)(77 87)(78 108)(80 95)(83 103)(85 90)(88 98)(91 106)(96 101)(99 109)(111 211)(112 177)(113 198)(114 219)(115 185)(116 206)(117 172)(118 193)(119 214)(120 180)(121 201)(122 167)(123 188)(124 209)(125 175)(126 196)(127 217)(128 183)(129 204)(130 170)(131 191)(132 212)(133 178)(134 199)(135 220)(136 186)(137 207)(138 173)(139 194)(140 215)(141 181)(142 202)(143 168)(144 189)(145 210)(146 176)(147 197)(148 218)(149 184)(150 205)(151 171)(152 192)(153 213)(154 179)(155 200)(156 166)(157 187)(158 208)(159 174)(160 195)(161 216)(162 182)(163 203)(164 169)(165 190)
G:=sub<Sym(220)| (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), (1,165,104,190)(2,164,105,189)(3,163,106,188)(4,162,107,187)(5,161,108,186)(6,160,109,185)(7,159,110,184)(8,158,56,183)(9,157,57,182)(10,156,58,181)(11,155,59,180)(12,154,60,179)(13,153,61,178)(14,152,62,177)(15,151,63,176)(16,150,64,175)(17,149,65,174)(18,148,66,173)(19,147,67,172)(20,146,68,171)(21,145,69,170)(22,144,70,169)(23,143,71,168)(24,142,72,167)(25,141,73,166)(26,140,74,220)(27,139,75,219)(28,138,76,218)(29,137,77,217)(30,136,78,216)(31,135,79,215)(32,134,80,214)(33,133,81,213)(34,132,82,212)(35,131,83,211)(36,130,84,210)(37,129,85,209)(38,128,86,208)(39,127,87,207)(40,126,88,206)(41,125,89,205)(42,124,90,204)(43,123,91,203)(44,122,92,202)(45,121,93,201)(46,120,94,200)(47,119,95,199)(48,118,96,198)(49,117,97,197)(50,116,98,196)(51,115,99,195)(52,114,100,194)(53,113,101,193)(54,112,102,192)(55,111,103,191), (2,22)(3,43)(4,9)(5,30)(6,51)(7,17)(8,38)(10,25)(11,46)(13,33)(14,54)(15,20)(16,41)(18,28)(19,49)(21,36)(24,44)(26,31)(27,52)(29,39)(32,47)(35,55)(37,42)(40,50)(48,53)(56,86)(57,107)(58,73)(59,94)(61,81)(62,102)(63,68)(64,89)(65,110)(66,76)(67,97)(69,84)(70,105)(72,92)(74,79)(75,100)(77,87)(78,108)(80,95)(83,103)(85,90)(88,98)(91,106)(96,101)(99,109)(111,211)(112,177)(113,198)(114,219)(115,185)(116,206)(117,172)(118,193)(119,214)(120,180)(121,201)(122,167)(123,188)(124,209)(125,175)(126,196)(127,217)(128,183)(129,204)(130,170)(131,191)(132,212)(133,178)(134,199)(135,220)(136,186)(137,207)(138,173)(139,194)(140,215)(141,181)(142,202)(143,168)(144,189)(145,210)(146,176)(147,197)(148,218)(149,184)(150,205)(151,171)(152,192)(153,213)(154,179)(155,200)(156,166)(157,187)(158,208)(159,174)(160,195)(161,216)(162,182)(163,203)(164,169)(165,190)>;
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), (1,165,104,190)(2,164,105,189)(3,163,106,188)(4,162,107,187)(5,161,108,186)(6,160,109,185)(7,159,110,184)(8,158,56,183)(9,157,57,182)(10,156,58,181)(11,155,59,180)(12,154,60,179)(13,153,61,178)(14,152,62,177)(15,151,63,176)(16,150,64,175)(17,149,65,174)(18,148,66,173)(19,147,67,172)(20,146,68,171)(21,145,69,170)(22,144,70,169)(23,143,71,168)(24,142,72,167)(25,141,73,166)(26,140,74,220)(27,139,75,219)(28,138,76,218)(29,137,77,217)(30,136,78,216)(31,135,79,215)(32,134,80,214)(33,133,81,213)(34,132,82,212)(35,131,83,211)(36,130,84,210)(37,129,85,209)(38,128,86,208)(39,127,87,207)(40,126,88,206)(41,125,89,205)(42,124,90,204)(43,123,91,203)(44,122,92,202)(45,121,93,201)(46,120,94,200)(47,119,95,199)(48,118,96,198)(49,117,97,197)(50,116,98,196)(51,115,99,195)(52,114,100,194)(53,113,101,193)(54,112,102,192)(55,111,103,191), (2,22)(3,43)(4,9)(5,30)(6,51)(7,17)(8,38)(10,25)(11,46)(13,33)(14,54)(15,20)(16,41)(18,28)(19,49)(21,36)(24,44)(26,31)(27,52)(29,39)(32,47)(35,55)(37,42)(40,50)(48,53)(56,86)(57,107)(58,73)(59,94)(61,81)(62,102)(63,68)(64,89)(65,110)(66,76)(67,97)(69,84)(70,105)(72,92)(74,79)(75,100)(77,87)(78,108)(80,95)(83,103)(85,90)(88,98)(91,106)(96,101)(99,109)(111,211)(112,177)(113,198)(114,219)(115,185)(116,206)(117,172)(118,193)(119,214)(120,180)(121,201)(122,167)(123,188)(124,209)(125,175)(126,196)(127,217)(128,183)(129,204)(130,170)(131,191)(132,212)(133,178)(134,199)(135,220)(136,186)(137,207)(138,173)(139,194)(140,215)(141,181)(142,202)(143,168)(144,189)(145,210)(146,176)(147,197)(148,218)(149,184)(150,205)(151,171)(152,192)(153,213)(154,179)(155,200)(156,166)(157,187)(158,208)(159,174)(160,195)(161,216)(162,182)(163,203)(164,169)(165,190) );
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)], [(1,165,104,190),(2,164,105,189),(3,163,106,188),(4,162,107,187),(5,161,108,186),(6,160,109,185),(7,159,110,184),(8,158,56,183),(9,157,57,182),(10,156,58,181),(11,155,59,180),(12,154,60,179),(13,153,61,178),(14,152,62,177),(15,151,63,176),(16,150,64,175),(17,149,65,174),(18,148,66,173),(19,147,67,172),(20,146,68,171),(21,145,69,170),(22,144,70,169),(23,143,71,168),(24,142,72,167),(25,141,73,166),(26,140,74,220),(27,139,75,219),(28,138,76,218),(29,137,77,217),(30,136,78,216),(31,135,79,215),(32,134,80,214),(33,133,81,213),(34,132,82,212),(35,131,83,211),(36,130,84,210),(37,129,85,209),(38,128,86,208),(39,127,87,207),(40,126,88,206),(41,125,89,205),(42,124,90,204),(43,123,91,203),(44,122,92,202),(45,121,93,201),(46,120,94,200),(47,119,95,199),(48,118,96,198),(49,117,97,197),(50,116,98,196),(51,115,99,195),(52,114,100,194),(53,113,101,193),(54,112,102,192),(55,111,103,191)], [(2,22),(3,43),(4,9),(5,30),(6,51),(7,17),(8,38),(10,25),(11,46),(13,33),(14,54),(15,20),(16,41),(18,28),(19,49),(21,36),(24,44),(26,31),(27,52),(29,39),(32,47),(35,55),(37,42),(40,50),(48,53),(56,86),(57,107),(58,73),(59,94),(61,81),(62,102),(63,68),(64,89),(65,110),(66,76),(67,97),(69,84),(70,105),(72,92),(74,79),(75,100),(77,87),(78,108),(80,95),(83,103),(85,90),(88,98),(91,106),(96,101),(99,109),(111,211),(112,177),(113,198),(114,219),(115,185),(116,206),(117,172),(118,193),(119,214),(120,180),(121,201),(122,167),(123,188),(124,209),(125,175),(126,196),(127,217),(128,183),(129,204),(130,170),(131,191),(132,212),(133,178),(134,199),(135,220),(136,186),(137,207),(138,173),(139,194),(140,215),(141,181),(142,202),(143,168),(144,189),(145,210),(146,176),(147,197),(148,218),(149,184),(150,205),(151,171),(152,192),(153,213),(154,179),(155,200),(156,166),(157,187),(158,208),(159,174),(160,195),(161,216),(162,182),(163,203),(164,169),(165,190)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 10A | 10B | 10C | 10D | 10E | 10F | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22O | 55A | ··· | 55J | 110A | ··· | 110J |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 10 | 22 | 110 | 2 | 2 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D10 | D11 | C5⋊D4 | D22 | C11⋊D4 | D5×D11 | C55⋊D4 |
| kernel | C55⋊D4 | Dic55 | C10×D11 | D5×C22 | C55 | D22 | C22 | D10 | C11 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 4 | 5 | 10 | 10 | 10 |
Matrix representation of C55⋊D4 ►in GL4(𝔽661) generated by
| 660 | 444 | 0 | 0 |
| 217 | 157 | 0 | 0 |
| 0 | 0 | 356 | 177 |
| 0 | 0 | 325 | 247 |
| 497 | 272 | 0 | 0 |
| 166 | 164 | 0 | 0 |
| 0 | 0 | 401 | 31 |
| 0 | 0 | 570 | 260 |
| 1 | 217 | 0 | 0 |
| 0 | 660 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(661))| [660,217,0,0,444,157,0,0,0,0,356,325,0,0,177,247],[497,166,0,0,272,164,0,0,0,0,401,570,0,0,31,260],[1,0,0,0,217,660,0,0,0,0,1,0,0,0,0,1] >;
C55⋊D4 in GAP, Magma, Sage, TeX
C_{55}\rtimes D_4 % in TeX
G:=Group("C55:D4"); // GroupNames label
G:=SmallGroup(440,20);
// by ID
G=gap.SmallGroup(440,20);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,61,328,10004]);
// Polycyclic
G:=Group<a,b,c|a^55=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^21,c*b*c=b^-1>;
// generators/relations
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