metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C55⋊7D4, C22⋊D55, D110⋊2C2, C2.5D110, Dic55⋊1C2, C22.12D10, C10.12D22, C110.12C22, (C2×C22)⋊2D5, (C2×C110)⋊2C2, (C2×C10)⋊2D11, C11⋊3(C5⋊D4), C5⋊3(C11⋊D4), SmallGroup(440,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C55⋊7D4
G = < a,b,c | a55=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C55⋊7D4
►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)
(1 133 95 195)(2 132 96 194)(3 131 97 193)(4 130 98 192)(5 129 99 191)(6 128 100 190)(7 127 101 189)(8 126 102 188)(9 125 103 187)(10 124 104 186)(11 123 105 185)(12 122 106 184)(13 121 107 183)(14 120 108 182)(15 119 109 181)(16 118 110 180)(17 117 56 179)(18 116 57 178)(19 115 58 177)(20 114 59 176)(21 113 60 175)(22 112 61 174)(23 111 62 173)(24 165 63 172)(25 164 64 171)(26 163 65 170)(27 162 66 169)(28 161 67 168)(29 160 68 167)(30 159 69 166)(31 158 70 220)(32 157 71 219)(33 156 72 218)(34 155 73 217)(35 154 74 216)(36 153 75 215)(37 152 76 214)(38 151 77 213)(39 150 78 212)(40 149 79 211)(41 148 80 210)(42 147 81 209)(43 146 82 208)(44 145 83 207)(45 144 84 206)(46 143 85 205)(47 142 86 204)(48 141 87 203)(49 140 88 202)(50 139 89 201)(51 138 90 200)(52 137 91 199)(53 136 92 198)(54 135 93 197)(55 134 94 196)
(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(56 79)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(80 110)(81 109)(82 108)(83 107)(84 106)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(111 217)(112 216)(113 215)(114 214)(115 213)(116 212)(117 211)(118 210)(119 209)(120 208)(121 207)(122 206)(123 205)(124 204)(125 203)(126 202)(127 201)(128 200)(129 199)(130 198)(131 197)(132 196)(133 195)(134 194)(135 193)(136 192)(137 191)(138 190)(139 189)(140 188)(141 187)(142 186)(143 185)(144 184)(145 183)(146 182)(147 181)(148 180)(149 179)(150 178)(151 177)(152 176)(153 175)(154 174)(155 173)(156 172)(157 171)(158 170)(159 169)(160 168)(161 167)(162 166)(163 220)(164 219)(165 218)
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), (1,133,95,195)(2,132,96,194)(3,131,97,193)(4,130,98,192)(5,129,99,191)(6,128,100,190)(7,127,101,189)(8,126,102,188)(9,125,103,187)(10,124,104,186)(11,123,105,185)(12,122,106,184)(13,121,107,183)(14,120,108,182)(15,119,109,181)(16,118,110,180)(17,117,56,179)(18,116,57,178)(19,115,58,177)(20,114,59,176)(21,113,60,175)(22,112,61,174)(23,111,62,173)(24,165,63,172)(25,164,64,171)(26,163,65,170)(27,162,66,169)(28,161,67,168)(29,160,68,167)(30,159,69,166)(31,158,70,220)(32,157,71,219)(33,156,72,218)(34,155,73,217)(35,154,74,216)(36,153,75,215)(37,152,76,214)(38,151,77,213)(39,150,78,212)(40,149,79,211)(41,148,80,210)(42,147,81,209)(43,146,82,208)(44,145,83,207)(45,144,84,206)(46,143,85,205)(47,142,86,204)(48,141,87,203)(49,140,88,202)(50,139,89,201)(51,138,90,200)(52,137,91,199)(53,136,92,198)(54,135,93,197)(55,134,94,196), (2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(80,110)(81,109)(82,108)(83,107)(84,106)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(111,217)(112,216)(113,215)(114,214)(115,213)(116,212)(117,211)(118,210)(119,209)(120,208)(121,207)(122,206)(123,205)(124,204)(125,203)(126,202)(127,201)(128,200)(129,199)(130,198)(131,197)(132,196)(133,195)(134,194)(135,193)(136,192)(137,191)(138,190)(139,189)(140,188)(141,187)(142,186)(143,185)(144,184)(145,183)(146,182)(147,181)(148,180)(149,179)(150,178)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,171)(158,170)(159,169)(160,168)(161,167)(162,166)(163,220)(164,219)(165,218)>;
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), (1,133,95,195)(2,132,96,194)(3,131,97,193)(4,130,98,192)(5,129,99,191)(6,128,100,190)(7,127,101,189)(8,126,102,188)(9,125,103,187)(10,124,104,186)(11,123,105,185)(12,122,106,184)(13,121,107,183)(14,120,108,182)(15,119,109,181)(16,118,110,180)(17,117,56,179)(18,116,57,178)(19,115,58,177)(20,114,59,176)(21,113,60,175)(22,112,61,174)(23,111,62,173)(24,165,63,172)(25,164,64,171)(26,163,65,170)(27,162,66,169)(28,161,67,168)(29,160,68,167)(30,159,69,166)(31,158,70,220)(32,157,71,219)(33,156,72,218)(34,155,73,217)(35,154,74,216)(36,153,75,215)(37,152,76,214)(38,151,77,213)(39,150,78,212)(40,149,79,211)(41,148,80,210)(42,147,81,209)(43,146,82,208)(44,145,83,207)(45,144,84,206)(46,143,85,205)(47,142,86,204)(48,141,87,203)(49,140,88,202)(50,139,89,201)(51,138,90,200)(52,137,91,199)(53,136,92,198)(54,135,93,197)(55,134,94,196), (2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(80,110)(81,109)(82,108)(83,107)(84,106)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,98)(93,97)(94,96)(111,217)(112,216)(113,215)(114,214)(115,213)(116,212)(117,211)(118,210)(119,209)(120,208)(121,207)(122,206)(123,205)(124,204)(125,203)(126,202)(127,201)(128,200)(129,199)(130,198)(131,197)(132,196)(133,195)(134,194)(135,193)(136,192)(137,191)(138,190)(139,189)(140,188)(141,187)(142,186)(143,185)(144,184)(145,183)(146,182)(147,181)(148,180)(149,179)(150,178)(151,177)(152,176)(153,175)(154,174)(155,173)(156,172)(157,171)(158,170)(159,169)(160,168)(161,167)(162,166)(163,220)(164,219)(165,218) );
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)], [(1,133,95,195),(2,132,96,194),(3,131,97,193),(4,130,98,192),(5,129,99,191),(6,128,100,190),(7,127,101,189),(8,126,102,188),(9,125,103,187),(10,124,104,186),(11,123,105,185),(12,122,106,184),(13,121,107,183),(14,120,108,182),(15,119,109,181),(16,118,110,180),(17,117,56,179),(18,116,57,178),(19,115,58,177),(20,114,59,176),(21,113,60,175),(22,112,61,174),(23,111,62,173),(24,165,63,172),(25,164,64,171),(26,163,65,170),(27,162,66,169),(28,161,67,168),(29,160,68,167),(30,159,69,166),(31,158,70,220),(32,157,71,219),(33,156,72,218),(34,155,73,217),(35,154,74,216),(36,153,75,215),(37,152,76,214),(38,151,77,213),(39,150,78,212),(40,149,79,211),(41,148,80,210),(42,147,81,209),(43,146,82,208),(44,145,83,207),(45,144,84,206),(46,143,85,205),(47,142,86,204),(48,141,87,203),(49,140,88,202),(50,139,89,201),(51,138,90,200),(52,137,91,199),(53,136,92,198),(54,135,93,197),(55,134,94,196)], [(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(56,79),(57,78),(58,77),(59,76),(60,75),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(80,110),(81,109),(82,108),(83,107),(84,106),(85,105),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,98),(93,97),(94,96),(111,217),(112,216),(113,215),(114,214),(115,213),(116,212),(117,211),(118,210),(119,209),(120,208),(121,207),(122,206),(123,205),(124,204),(125,203),(126,202),(127,201),(128,200),(129,199),(130,198),(131,197),(132,196),(133,195),(134,194),(135,193),(136,192),(137,191),(138,190),(139,189),(140,188),(141,187),(142,186),(143,185),(144,184),(145,183),(146,182),(147,181),(148,180),(149,179),(150,178),(151,177),(152,176),(153,175),(154,174),(155,173),(156,172),(157,171),(158,170),(159,169),(160,168),(161,167),(162,166),(163,220),(164,219),(165,218)]])
113 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 10A | ··· | 10F | 11A | ··· | 11E | 22A | ··· | 22O | 55A | ··· | 55T | 110A | ··· | 110BH |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 10 | ··· | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 2 | 110 | 110 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
113 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | D4 | D5 | D10 | D11 | C5⋊D4 | D22 | C11⋊D4 | D55 | D110 | C55⋊7D4 |
| kernel | C55⋊7D4 | Dic55 | D110 | C2×C110 | C55 | C2×C22 | C22 | C2×C10 | C11 | C10 | C5 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 4 | 5 | 10 | 20 | 20 | 40 |
Matrix representation of C55⋊7D4 ►in GL2(𝔽661) generated by
| 69 | 482 |
| 357 | 530 |
| 614 | 355 |
| 301 | 47 |
| 605 | 19 |
| 496 | 56 |
G:=sub<GL(2,GF(661))| [69,357,482,530],[614,301,355,47],[605,496,19,56] >;
C55⋊7D4 in GAP, Magma, Sage, TeX
C_{55}\rtimes_7D_4 % in TeX
G:=Group("C55:7D4"); // GroupNames label
G:=SmallGroup(440,38);
// by ID
G=gap.SmallGroup(440,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,61,643,10004]);
// Polycyclic
G:=Group<a,b,c|a^55=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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