direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22⋊C16, C22⋊C112, C56.106D4, C23.2C56, C14.8M5(2), C28.37M4(2), (C2×C14)⋊1C16, (C2×C112)⋊5C2, (C2×C16)⋊1C14, (C2×C8).6C28, (C2×C4).3C56, (C2×C28).7C8, C8.26(C7×D4), (C2×C56).16C4, C2.1(C2×C112), C14.11(C2×C16), (C22×C8).4C14, C22.9(C2×C56), (C22×C4).8C28, (C22×C56).7C2, (C22×C14).3C8, C2.2(C7×M5(2)), (C22×C28).15C4, C4.10(C7×M4(2)), C14.25(C22⋊C8), (C2×C56).451C22, C28.111(C22⋊C4), C2.2(C7×C22⋊C8), (C2×C4).83(C2×C28), (C2×C14).40(C2×C8), C4.28(C7×C22⋊C4), (C2×C28).345(C2×C4), (C2×C8).105(C2×C14), SmallGroup(448,152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22⋊C16
G = < a,b,c,d | a7=b2=c2=d16=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 90 in 66 conjugacy classes, 42 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C16, C2×C8, C2×C8, C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C2×C16, C22×C8, C56, C56, C2×C28, C2×C28, C22×C14, C22⋊C16, C112, C2×C56, C2×C56, C22×C28, C2×C112, C22×C56, C7×C22⋊C16
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, D4, C14, C16, C22⋊C4, C2×C8, M4(2), C28, C2×C14, C22⋊C8, C2×C16, M5(2), C56, C2×C28, C7×D4, C22⋊C16, C112, C7×C22⋊C4, C2×C56, C7×M4(2), C7×C22⋊C8, C2×C112, C7×M5(2), C7×C22⋊C16
►On 224 points
Generators in S224
(1 163 93 119 50 111 191)(2 164 94 120 51 112 192)(3 165 95 121 52 97 177)(4 166 96 122 53 98 178)(5 167 81 123 54 99 179)(6 168 82 124 55 100 180)(7 169 83 125 56 101 181)(8 170 84 126 57 102 182)(9 171 85 127 58 103 183)(10 172 86 128 59 104 184)(11 173 87 113 60 105 185)(12 174 88 114 61 106 186)(13 175 89 115 62 107 187)(14 176 90 116 63 108 188)(15 161 91 117 64 109 189)(16 162 92 118 49 110 190)(17 66 150 197 133 216 36)(18 67 151 198 134 217 37)(19 68 152 199 135 218 38)(20 69 153 200 136 219 39)(21 70 154 201 137 220 40)(22 71 155 202 138 221 41)(23 72 156 203 139 222 42)(24 73 157 204 140 223 43)(25 74 158 205 141 224 44)(26 75 159 206 142 209 45)(27 76 160 207 143 210 46)(28 77 145 208 144 211 47)(29 78 146 193 129 212 48)(30 79 147 194 130 213 33)(31 80 148 195 131 214 34)(32 65 149 196 132 215 35)
(2 78)(4 80)(6 66)(8 68)(10 70)(12 72)(14 74)(16 76)(17 180)(19 182)(21 184)(23 186)(25 188)(27 190)(29 192)(31 178)(34 98)(36 100)(38 102)(40 104)(42 106)(44 108)(46 110)(48 112)(49 210)(51 212)(53 214)(55 216)(57 218)(59 220)(61 222)(63 224)(82 197)(84 199)(86 201)(88 203)(90 205)(92 207)(94 193)(96 195)(114 139)(116 141)(118 143)(120 129)(122 131)(124 133)(126 135)(128 137)(146 164)(148 166)(150 168)(152 170)(154 172)(156 174)(158 176)(160 162)
(1 77)(2 78)(3 79)(4 80)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 180)(18 181)(19 182)(20 183)(21 184)(22 185)(23 186)(24 187)(25 188)(26 189)(27 190)(28 191)(29 192)(30 177)(31 178)(32 179)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 210)(50 211)(51 212)(52 213)(53 214)(54 215)(55 216)(56 217)(57 218)(58 219)(59 220)(60 221)(61 222)(62 223)(63 224)(64 209)(81 196)(82 197)(83 198)(84 199)(85 200)(86 201)(87 202)(88 203)(89 204)(90 205)(91 206)(92 207)(93 208)(94 193)(95 194)(96 195)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 129)(121 130)(122 131)(123 132)(124 133)(125 134)(126 135)(127 136)(128 137)(145 163)(146 164)(147 165)(148 166)(149 167)(150 168)(151 169)(152 170)(153 171)(154 172)(155 173)(156 174)(157 175)(158 176)(159 161)(160 162)
(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)
G:=sub<Sym(224)| (1,163,93,119,50,111,191)(2,164,94,120,51,112,192)(3,165,95,121,52,97,177)(4,166,96,122,53,98,178)(5,167,81,123,54,99,179)(6,168,82,124,55,100,180)(7,169,83,125,56,101,181)(8,170,84,126,57,102,182)(9,171,85,127,58,103,183)(10,172,86,128,59,104,184)(11,173,87,113,60,105,185)(12,174,88,114,61,106,186)(13,175,89,115,62,107,187)(14,176,90,116,63,108,188)(15,161,91,117,64,109,189)(16,162,92,118,49,110,190)(17,66,150,197,133,216,36)(18,67,151,198,134,217,37)(19,68,152,199,135,218,38)(20,69,153,200,136,219,39)(21,70,154,201,137,220,40)(22,71,155,202,138,221,41)(23,72,156,203,139,222,42)(24,73,157,204,140,223,43)(25,74,158,205,141,224,44)(26,75,159,206,142,209,45)(27,76,160,207,143,210,46)(28,77,145,208,144,211,47)(29,78,146,193,129,212,48)(30,79,147,194,130,213,33)(31,80,148,195,131,214,34)(32,65,149,196,132,215,35), (2,78)(4,80)(6,66)(8,68)(10,70)(12,72)(14,74)(16,76)(17,180)(19,182)(21,184)(23,186)(25,188)(27,190)(29,192)(31,178)(34,98)(36,100)(38,102)(40,104)(42,106)(44,108)(46,110)(48,112)(49,210)(51,212)(53,214)(55,216)(57,218)(59,220)(61,222)(63,224)(82,197)(84,199)(86,201)(88,203)(90,205)(92,207)(94,193)(96,195)(114,139)(116,141)(118,143)(120,129)(122,131)(124,133)(126,135)(128,137)(146,164)(148,166)(150,168)(152,170)(154,172)(156,174)(158,176)(160,162), (1,77)(2,78)(3,79)(4,80)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,180)(18,181)(19,182)(20,183)(21,184)(22,185)(23,186)(24,187)(25,188)(26,189)(27,190)(28,191)(29,192)(30,177)(31,178)(32,179)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,210)(50,211)(51,212)(52,213)(53,214)(54,215)(55,216)(56,217)(57,218)(58,219)(59,220)(60,221)(61,222)(62,223)(63,224)(64,209)(81,196)(82,197)(83,198)(84,199)(85,200)(86,201)(87,202)(88,203)(89,204)(90,205)(91,206)(92,207)(93,208)(94,193)(95,194)(96,195)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(127,136)(128,137)(145,163)(146,164)(147,165)(148,166)(149,167)(150,168)(151,169)(152,170)(153,171)(154,172)(155,173)(156,174)(157,175)(158,176)(159,161)(160,162), (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)>;
G:=Group( (1,163,93,119,50,111,191)(2,164,94,120,51,112,192)(3,165,95,121,52,97,177)(4,166,96,122,53,98,178)(5,167,81,123,54,99,179)(6,168,82,124,55,100,180)(7,169,83,125,56,101,181)(8,170,84,126,57,102,182)(9,171,85,127,58,103,183)(10,172,86,128,59,104,184)(11,173,87,113,60,105,185)(12,174,88,114,61,106,186)(13,175,89,115,62,107,187)(14,176,90,116,63,108,188)(15,161,91,117,64,109,189)(16,162,92,118,49,110,190)(17,66,150,197,133,216,36)(18,67,151,198,134,217,37)(19,68,152,199,135,218,38)(20,69,153,200,136,219,39)(21,70,154,201,137,220,40)(22,71,155,202,138,221,41)(23,72,156,203,139,222,42)(24,73,157,204,140,223,43)(25,74,158,205,141,224,44)(26,75,159,206,142,209,45)(27,76,160,207,143,210,46)(28,77,145,208,144,211,47)(29,78,146,193,129,212,48)(30,79,147,194,130,213,33)(31,80,148,195,131,214,34)(32,65,149,196,132,215,35), (2,78)(4,80)(6,66)(8,68)(10,70)(12,72)(14,74)(16,76)(17,180)(19,182)(21,184)(23,186)(25,188)(27,190)(29,192)(31,178)(34,98)(36,100)(38,102)(40,104)(42,106)(44,108)(46,110)(48,112)(49,210)(51,212)(53,214)(55,216)(57,218)(59,220)(61,222)(63,224)(82,197)(84,199)(86,201)(88,203)(90,205)(92,207)(94,193)(96,195)(114,139)(116,141)(118,143)(120,129)(122,131)(124,133)(126,135)(128,137)(146,164)(148,166)(150,168)(152,170)(154,172)(156,174)(158,176)(160,162), (1,77)(2,78)(3,79)(4,80)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,180)(18,181)(19,182)(20,183)(21,184)(22,185)(23,186)(24,187)(25,188)(26,189)(27,190)(28,191)(29,192)(30,177)(31,178)(32,179)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,210)(50,211)(51,212)(52,213)(53,214)(54,215)(55,216)(56,217)(57,218)(58,219)(59,220)(60,221)(61,222)(62,223)(63,224)(64,209)(81,196)(82,197)(83,198)(84,199)(85,200)(86,201)(87,202)(88,203)(89,204)(90,205)(91,206)(92,207)(93,208)(94,193)(95,194)(96,195)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(127,136)(128,137)(145,163)(146,164)(147,165)(148,166)(149,167)(150,168)(151,169)(152,170)(153,171)(154,172)(155,173)(156,174)(157,175)(158,176)(159,161)(160,162), (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) );
G=PermutationGroup([[(1,163,93,119,50,111,191),(2,164,94,120,51,112,192),(3,165,95,121,52,97,177),(4,166,96,122,53,98,178),(5,167,81,123,54,99,179),(6,168,82,124,55,100,180),(7,169,83,125,56,101,181),(8,170,84,126,57,102,182),(9,171,85,127,58,103,183),(10,172,86,128,59,104,184),(11,173,87,113,60,105,185),(12,174,88,114,61,106,186),(13,175,89,115,62,107,187),(14,176,90,116,63,108,188),(15,161,91,117,64,109,189),(16,162,92,118,49,110,190),(17,66,150,197,133,216,36),(18,67,151,198,134,217,37),(19,68,152,199,135,218,38),(20,69,153,200,136,219,39),(21,70,154,201,137,220,40),(22,71,155,202,138,221,41),(23,72,156,203,139,222,42),(24,73,157,204,140,223,43),(25,74,158,205,141,224,44),(26,75,159,206,142,209,45),(27,76,160,207,143,210,46),(28,77,145,208,144,211,47),(29,78,146,193,129,212,48),(30,79,147,194,130,213,33),(31,80,148,195,131,214,34),(32,65,149,196,132,215,35)], [(2,78),(4,80),(6,66),(8,68),(10,70),(12,72),(14,74),(16,76),(17,180),(19,182),(21,184),(23,186),(25,188),(27,190),(29,192),(31,178),(34,98),(36,100),(38,102),(40,104),(42,106),(44,108),(46,110),(48,112),(49,210),(51,212),(53,214),(55,216),(57,218),(59,220),(61,222),(63,224),(82,197),(84,199),(86,201),(88,203),(90,205),(92,207),(94,193),(96,195),(114,139),(116,141),(118,143),(120,129),(122,131),(124,133),(126,135),(128,137),(146,164),(148,166),(150,168),(152,170),(154,172),(156,174),(158,176),(160,162)], [(1,77),(2,78),(3,79),(4,80),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,74),(15,75),(16,76),(17,180),(18,181),(19,182),(20,183),(21,184),(22,185),(23,186),(24,187),(25,188),(26,189),(27,190),(28,191),(29,192),(30,177),(31,178),(32,179),(33,97),(34,98),(35,99),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,210),(50,211),(51,212),(52,213),(53,214),(54,215),(55,216),(56,217),(57,218),(58,219),(59,220),(60,221),(61,222),(62,223),(63,224),(64,209),(81,196),(82,197),(83,198),(84,199),(85,200),(86,201),(87,202),(88,203),(89,204),(90,205),(91,206),(92,207),(93,208),(94,193),(95,194),(96,195),(113,138),(114,139),(115,140),(116,141),(117,142),(118,143),(119,144),(120,129),(121,130),(122,131),(123,132),(124,133),(125,134),(126,135),(127,136),(128,137),(145,163),(146,164),(147,165),(148,166),(149,167),(150,168),(151,169),(152,170),(153,171),(154,172),(155,173),(156,174),(157,175),(158,176),(159,161),(160,162)], [(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)]])
280 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14R | 14S | ··· | 14AD | 16A | ··· | 16P | 28A | ··· | 28X | 28Y | ··· | 28AJ | 56A | ··· | 56AV | 56AW | ··· | 56BT | 112A | ··· | 112CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C8 | C8 | C14 | C14 | C16 | C28 | C28 | C56 | C56 | C112 | D4 | M4(2) | M5(2) | C7×D4 | C7×M4(2) | C7×M5(2) |
| kernel | C7×C22⋊C16 | C2×C112 | C22×C56 | C2×C56 | C22×C28 | C22⋊C16 | C2×C28 | C22×C14 | C2×C16 | C22×C8 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C56 | C28 | C14 | C8 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 4 | 4 | 12 | 6 | 16 | 12 | 12 | 24 | 24 | 96 | 2 | 2 | 4 | 12 | 12 | 24 |
Matrix representation of C7×C22⋊C16 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 112 | 0 | 0 |
| 0 | 1 | 109 |
| 0 | 0 | 112 |
| 1 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 78 | 0 | 0 |
| 0 | 41 | 87 |
| 0 | 77 | 72 |
G:=sub<GL(3,GF(113))| [1,0,0,0,16,0,0,0,16],[112,0,0,0,1,0,0,109,112],[1,0,0,0,112,0,0,0,112],[78,0,0,0,41,77,0,87,72] >;
C7×C22⋊C16 in GAP, Magma, Sage, TeX
C_7\times C_2^2\rtimes C_{16} % in TeX
G:=Group("C7xC2^2:C16"); // GroupNames label
G:=SmallGroup(448,152);
// by ID
G=gap.SmallGroup(448,152);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations