metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.228D14, (C4×D7)⋊10D4, (C4×D4)⋊10D7, C28⋊Q8⋊47C2, (D4×C28)⋊12C2, (C4×D28)⋊26C2, C28⋊1(C4○D4), C4⋊2(C4○D28), D14.1(C2×D4), C4.220(D4×D7), (D7×C42)⋊4C2, C28⋊1D4⋊43C2, C28⋊D4⋊32C2, C28⋊2D4⋊45C2, C4⋊C4.282D14, C28.379(C2×D4), Dic7.3(C2×D4), D14⋊D4⋊48C2, Dic7⋊1(C4○D4), (C2×D4).212D14, (C2×C14).92C24, C14.48(C22×D4), (C4×C28).151C22, (C2×C28).492C23, C22⋊C4.109D14, (C22×C4).207D14, C23.93(C22×D7), D14⋊C4.122C22, Dic7.D4⋊50C2, (D4×C14).255C22, (C2×D28).259C22, C4⋊Dic7.363C22, C7⋊2(C22.26C24), (C22×D7).30C23, C22.117(C23×D7), Dic7⋊C4.110C22, (C22×C28).106C22, (C22×C14).162C23, (C2×Dic7).203C23, (C4×Dic7).251C22, C23.D7.105C22, (C2×Dic14).238C22, C2.20(C2×D4×D7), (C4×C7⋊D4)⋊4C2, (C2×C4○D28)⋊6C2, C2.21(D7×C4○D4), C14.40(C2×C4○D4), C2.44(C2×C4○D28), (C2×C4×D7).292C22, (C7×C4⋊C4).325C22, (C2×C4).578(C22×D7), (C2×C7⋊D4).112C22, (C7×C22⋊C4).121C22, SmallGroup(448,1001)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1492 in 310 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×10], C22, C22 [×16], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×21], D4 [×20], Q8 [×4], C23 [×2], C23 [×3], D7 [×4], C14 [×3], C14 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4 [×9], C2×Q8 [×2], C4○D4 [×8], Dic7 [×4], Dic7 [×3], C28 [×4], C28 [×3], D14 [×2], D14 [×8], C2×C14, C2×C14 [×6], C2×C42, C4×D4, C4×D4 [×3], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×C4○D4 [×2], Dic14 [×4], C4×D7 [×4], C4×D7 [×8], D28 [×6], C2×Dic7 [×3], C2×Dic7 [×2], C7⋊D4 [×12], C2×C28 [×3], C2×C28 [×2], C2×C28 [×4], C7×D4 [×2], C22×D7, C22×D7 [×2], C22×C14 [×2], C22.26C24, C4×Dic7 [×3], Dic7⋊C4 [×2], C4⋊Dic7, D14⋊C4 [×4], C23.D7 [×2], C4×C28, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7 [×3], C2×C4×D7 [×2], C2×D28, C2×D28 [×2], C4○D28 [×8], C2×C7⋊D4 [×6], C22×C28 [×2], D4×C14, D7×C42, C4×D28, D14⋊D4 [×2], Dic7.D4 [×2], C28⋊Q8, C28⋊1D4, C4×C7⋊D4 [×2], C28⋊2D4, C28⋊D4, D4×C28, C2×C4○D28 [×2], C42.228D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×4], C24, D14 [×7], C22×D4, C2×C4○D4 [×2], C22×D7 [×7], C22.26C24, C4○D28 [×2], D4×D7 [×2], C23×D7, C2×C4○D28, C2×D4×D7, D7×C4○D4, C42.228D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=1, d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >
►On 224 points
Generators in S224
(1 92 56 145)(2 146 43 93)(3 94 44 147)(4 148 45 95)(5 96 46 149)(6 150 47 97)(7 98 48 151)(8 152 49 85)(9 86 50 153)(10 154 51 87)(11 88 52 141)(12 142 53 89)(13 90 54 143)(14 144 55 91)(15 217 192 197)(16 198 193 218)(17 219 194 199)(18 200 195 220)(19 221 196 201)(20 202 183 222)(21 223 184 203)(22 204 185 224)(23 211 186 205)(24 206 187 212)(25 213 188 207)(26 208 189 214)(27 215 190 209)(28 210 191 216)(29 57 101 159)(30 160 102 58)(31 59 103 161)(32 162 104 60)(33 61 105 163)(34 164 106 62)(35 63 107 165)(36 166 108 64)(37 65 109 167)(38 168 110 66)(39 67 111 155)(40 156 112 68)(41 69 99 157)(42 158 100 70)(71 118 132 170)(72 171 133 119)(73 120 134 172)(74 173 135 121)(75 122 136 174)(76 175 137 123)(77 124 138 176)(78 177 139 125)(79 126 140 178)(80 179 127 113)(81 114 128 180)(82 181 129 115)(83 116 130 182)(84 169 131 117)
(1 115 163 188)(2 116 164 189)(3 117 165 190)(4 118 166 191)(5 119 167 192)(6 120 168 193)(7 121 155 194)(8 122 156 195)(9 123 157 196)(10 124 158 183)(11 125 159 184)(12 126 160 185)(13 113 161 186)(14 114 162 187)(15 46 171 65)(16 47 172 66)(17 48 173 67)(18 49 174 68)(19 50 175 69)(20 51 176 70)(21 52 177 57)(22 53 178 58)(23 54 179 59)(24 55 180 60)(25 56 181 61)(26 43 182 62)(27 44 169 63)(28 45 170 64)(29 203 88 78)(30 204 89 79)(31 205 90 80)(32 206 91 81)(33 207 92 82)(34 208 93 83)(35 209 94 84)(36 210 95 71)(37 197 96 72)(38 198 97 73)(39 199 98 74)(40 200 85 75)(41 201 86 76)(42 202 87 77)(99 221 153 137)(100 222 154 138)(101 223 141 139)(102 224 142 140)(103 211 143 127)(104 212 144 128)(105 213 145 129)(106 214 146 130)(107 215 147 131)(108 216 148 132)(109 217 149 133)(110 218 150 134)(111 219 151 135)(112 220 152 136)
(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)
(1 14 163 162)(2 161 164 13)(3 12 165 160)(4 159 166 11)(5 10 167 158)(6 157 168 9)(7 8 155 156)(15 20 171 176)(16 175 172 19)(17 18 173 174)(21 28 177 170)(22 169 178 27)(23 26 179 182)(24 181 180 25)(29 36 88 95)(30 94 89 35)(31 34 90 93)(32 92 91 33)(37 42 96 87)(38 86 97 41)(39 40 98 85)(43 59 62 54)(44 53 63 58)(45 57 64 52)(46 51 65 70)(47 69 66 50)(48 49 67 68)(55 61 60 56)(71 203 210 78)(72 77 197 202)(73 201 198 76)(74 75 199 200)(79 209 204 84)(80 83 205 208)(81 207 206 82)(99 110 153 150)(100 149 154 109)(101 108 141 148)(102 147 142 107)(103 106 143 146)(104 145 144 105)(111 112 151 152)(113 116 186 189)(114 188 187 115)(117 126 190 185)(118 184 191 125)(119 124 192 183)(120 196 193 123)(121 122 194 195)(127 130 211 214)(128 213 212 129)(131 140 215 224)(132 223 216 139)(133 138 217 222)(134 221 218 137)(135 136 219 220)
G:=sub<Sym(224)| (1,92,56,145)(2,146,43,93)(3,94,44,147)(4,148,45,95)(5,96,46,149)(6,150,47,97)(7,98,48,151)(8,152,49,85)(9,86,50,153)(10,154,51,87)(11,88,52,141)(12,142,53,89)(13,90,54,143)(14,144,55,91)(15,217,192,197)(16,198,193,218)(17,219,194,199)(18,200,195,220)(19,221,196,201)(20,202,183,222)(21,223,184,203)(22,204,185,224)(23,211,186,205)(24,206,187,212)(25,213,188,207)(26,208,189,214)(27,215,190,209)(28,210,191,216)(29,57,101,159)(30,160,102,58)(31,59,103,161)(32,162,104,60)(33,61,105,163)(34,164,106,62)(35,63,107,165)(36,166,108,64)(37,65,109,167)(38,168,110,66)(39,67,111,155)(40,156,112,68)(41,69,99,157)(42,158,100,70)(71,118,132,170)(72,171,133,119)(73,120,134,172)(74,173,135,121)(75,122,136,174)(76,175,137,123)(77,124,138,176)(78,177,139,125)(79,126,140,178)(80,179,127,113)(81,114,128,180)(82,181,129,115)(83,116,130,182)(84,169,131,117), (1,115,163,188)(2,116,164,189)(3,117,165,190)(4,118,166,191)(5,119,167,192)(6,120,168,193)(7,121,155,194)(8,122,156,195)(9,123,157,196)(10,124,158,183)(11,125,159,184)(12,126,160,185)(13,113,161,186)(14,114,162,187)(15,46,171,65)(16,47,172,66)(17,48,173,67)(18,49,174,68)(19,50,175,69)(20,51,176,70)(21,52,177,57)(22,53,178,58)(23,54,179,59)(24,55,180,60)(25,56,181,61)(26,43,182,62)(27,44,169,63)(28,45,170,64)(29,203,88,78)(30,204,89,79)(31,205,90,80)(32,206,91,81)(33,207,92,82)(34,208,93,83)(35,209,94,84)(36,210,95,71)(37,197,96,72)(38,198,97,73)(39,199,98,74)(40,200,85,75)(41,201,86,76)(42,202,87,77)(99,221,153,137)(100,222,154,138)(101,223,141,139)(102,224,142,140)(103,211,143,127)(104,212,144,128)(105,213,145,129)(106,214,146,130)(107,215,147,131)(108,216,148,132)(109,217,149,133)(110,218,150,134)(111,219,151,135)(112,220,152,136), (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), (1,14,163,162)(2,161,164,13)(3,12,165,160)(4,159,166,11)(5,10,167,158)(6,157,168,9)(7,8,155,156)(15,20,171,176)(16,175,172,19)(17,18,173,174)(21,28,177,170)(22,169,178,27)(23,26,179,182)(24,181,180,25)(29,36,88,95)(30,94,89,35)(31,34,90,93)(32,92,91,33)(37,42,96,87)(38,86,97,41)(39,40,98,85)(43,59,62,54)(44,53,63,58)(45,57,64,52)(46,51,65,70)(47,69,66,50)(48,49,67,68)(55,61,60,56)(71,203,210,78)(72,77,197,202)(73,201,198,76)(74,75,199,200)(79,209,204,84)(80,83,205,208)(81,207,206,82)(99,110,153,150)(100,149,154,109)(101,108,141,148)(102,147,142,107)(103,106,143,146)(104,145,144,105)(111,112,151,152)(113,116,186,189)(114,188,187,115)(117,126,190,185)(118,184,191,125)(119,124,192,183)(120,196,193,123)(121,122,194,195)(127,130,211,214)(128,213,212,129)(131,140,215,224)(132,223,216,139)(133,138,217,222)(134,221,218,137)(135,136,219,220)>;
G:=Group( (1,92,56,145)(2,146,43,93)(3,94,44,147)(4,148,45,95)(5,96,46,149)(6,150,47,97)(7,98,48,151)(8,152,49,85)(9,86,50,153)(10,154,51,87)(11,88,52,141)(12,142,53,89)(13,90,54,143)(14,144,55,91)(15,217,192,197)(16,198,193,218)(17,219,194,199)(18,200,195,220)(19,221,196,201)(20,202,183,222)(21,223,184,203)(22,204,185,224)(23,211,186,205)(24,206,187,212)(25,213,188,207)(26,208,189,214)(27,215,190,209)(28,210,191,216)(29,57,101,159)(30,160,102,58)(31,59,103,161)(32,162,104,60)(33,61,105,163)(34,164,106,62)(35,63,107,165)(36,166,108,64)(37,65,109,167)(38,168,110,66)(39,67,111,155)(40,156,112,68)(41,69,99,157)(42,158,100,70)(71,118,132,170)(72,171,133,119)(73,120,134,172)(74,173,135,121)(75,122,136,174)(76,175,137,123)(77,124,138,176)(78,177,139,125)(79,126,140,178)(80,179,127,113)(81,114,128,180)(82,181,129,115)(83,116,130,182)(84,169,131,117), (1,115,163,188)(2,116,164,189)(3,117,165,190)(4,118,166,191)(5,119,167,192)(6,120,168,193)(7,121,155,194)(8,122,156,195)(9,123,157,196)(10,124,158,183)(11,125,159,184)(12,126,160,185)(13,113,161,186)(14,114,162,187)(15,46,171,65)(16,47,172,66)(17,48,173,67)(18,49,174,68)(19,50,175,69)(20,51,176,70)(21,52,177,57)(22,53,178,58)(23,54,179,59)(24,55,180,60)(25,56,181,61)(26,43,182,62)(27,44,169,63)(28,45,170,64)(29,203,88,78)(30,204,89,79)(31,205,90,80)(32,206,91,81)(33,207,92,82)(34,208,93,83)(35,209,94,84)(36,210,95,71)(37,197,96,72)(38,198,97,73)(39,199,98,74)(40,200,85,75)(41,201,86,76)(42,202,87,77)(99,221,153,137)(100,222,154,138)(101,223,141,139)(102,224,142,140)(103,211,143,127)(104,212,144,128)(105,213,145,129)(106,214,146,130)(107,215,147,131)(108,216,148,132)(109,217,149,133)(110,218,150,134)(111,219,151,135)(112,220,152,136), (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), (1,14,163,162)(2,161,164,13)(3,12,165,160)(4,159,166,11)(5,10,167,158)(6,157,168,9)(7,8,155,156)(15,20,171,176)(16,175,172,19)(17,18,173,174)(21,28,177,170)(22,169,178,27)(23,26,179,182)(24,181,180,25)(29,36,88,95)(30,94,89,35)(31,34,90,93)(32,92,91,33)(37,42,96,87)(38,86,97,41)(39,40,98,85)(43,59,62,54)(44,53,63,58)(45,57,64,52)(46,51,65,70)(47,69,66,50)(48,49,67,68)(55,61,60,56)(71,203,210,78)(72,77,197,202)(73,201,198,76)(74,75,199,200)(79,209,204,84)(80,83,205,208)(81,207,206,82)(99,110,153,150)(100,149,154,109)(101,108,141,148)(102,147,142,107)(103,106,143,146)(104,145,144,105)(111,112,151,152)(113,116,186,189)(114,188,187,115)(117,126,190,185)(118,184,191,125)(119,124,192,183)(120,196,193,123)(121,122,194,195)(127,130,211,214)(128,213,212,129)(131,140,215,224)(132,223,216,139)(133,138,217,222)(134,221,218,137)(135,136,219,220) );
G=PermutationGroup([(1,92,56,145),(2,146,43,93),(3,94,44,147),(4,148,45,95),(5,96,46,149),(6,150,47,97),(7,98,48,151),(8,152,49,85),(9,86,50,153),(10,154,51,87),(11,88,52,141),(12,142,53,89),(13,90,54,143),(14,144,55,91),(15,217,192,197),(16,198,193,218),(17,219,194,199),(18,200,195,220),(19,221,196,201),(20,202,183,222),(21,223,184,203),(22,204,185,224),(23,211,186,205),(24,206,187,212),(25,213,188,207),(26,208,189,214),(27,215,190,209),(28,210,191,216),(29,57,101,159),(30,160,102,58),(31,59,103,161),(32,162,104,60),(33,61,105,163),(34,164,106,62),(35,63,107,165),(36,166,108,64),(37,65,109,167),(38,168,110,66),(39,67,111,155),(40,156,112,68),(41,69,99,157),(42,158,100,70),(71,118,132,170),(72,171,133,119),(73,120,134,172),(74,173,135,121),(75,122,136,174),(76,175,137,123),(77,124,138,176),(78,177,139,125),(79,126,140,178),(80,179,127,113),(81,114,128,180),(82,181,129,115),(83,116,130,182),(84,169,131,117)], [(1,115,163,188),(2,116,164,189),(3,117,165,190),(4,118,166,191),(5,119,167,192),(6,120,168,193),(7,121,155,194),(8,122,156,195),(9,123,157,196),(10,124,158,183),(11,125,159,184),(12,126,160,185),(13,113,161,186),(14,114,162,187),(15,46,171,65),(16,47,172,66),(17,48,173,67),(18,49,174,68),(19,50,175,69),(20,51,176,70),(21,52,177,57),(22,53,178,58),(23,54,179,59),(24,55,180,60),(25,56,181,61),(26,43,182,62),(27,44,169,63),(28,45,170,64),(29,203,88,78),(30,204,89,79),(31,205,90,80),(32,206,91,81),(33,207,92,82),(34,208,93,83),(35,209,94,84),(36,210,95,71),(37,197,96,72),(38,198,97,73),(39,199,98,74),(40,200,85,75),(41,201,86,76),(42,202,87,77),(99,221,153,137),(100,222,154,138),(101,223,141,139),(102,224,142,140),(103,211,143,127),(104,212,144,128),(105,213,145,129),(106,214,146,130),(107,215,147,131),(108,216,148,132),(109,217,149,133),(110,218,150,134),(111,219,151,135),(112,220,152,136)], [(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)], [(1,14,163,162),(2,161,164,13),(3,12,165,160),(4,159,166,11),(5,10,167,158),(6,157,168,9),(7,8,155,156),(15,20,171,176),(16,175,172,19),(17,18,173,174),(21,28,177,170),(22,169,178,27),(23,26,179,182),(24,181,180,25),(29,36,88,95),(30,94,89,35),(31,34,90,93),(32,92,91,33),(37,42,96,87),(38,86,97,41),(39,40,98,85),(43,59,62,54),(44,53,63,58),(45,57,64,52),(46,51,65,70),(47,69,66,50),(48,49,67,68),(55,61,60,56),(71,203,210,78),(72,77,197,202),(73,201,198,76),(74,75,199,200),(79,209,204,84),(80,83,205,208),(81,207,206,82),(99,110,153,150),(100,149,154,109),(101,108,141,148),(102,147,142,107),(103,106,143,146),(104,145,144,105),(111,112,151,152),(113,116,186,189),(114,188,187,115),(117,126,190,185),(118,184,191,125),(119,124,192,183),(120,196,193,123),(121,122,194,195),(127,130,211,214),(128,213,212,129),(131,140,215,224),(132,223,216,139),(133,138,217,222),(134,221,218,137),(135,136,219,220)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 17 | 12 |
| 17 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
| 22 | 7 | 0 | 0 |
| 22 | 3 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 28 |
| 7 | 22 | 0 | 0 |
| 3 | 22 | 0 | 0 |
| 0 | 0 | 28 | 2 |
| 0 | 0 | 28 | 1 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,17,17,0,0,0,12],[17,0,0,0,0,17,0,0,0,0,17,0,0,0,0,17],[22,22,0,0,7,3,0,0,0,0,1,0,0,0,27,28],[7,3,0,0,22,22,0,0,0,0,28,28,0,0,2,1] >;
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | D14 | C4○D28 | D4×D7 | D7×C4○D4 |
| kernel | C42.228D14 | D7×C42 | C4×D28 | D14⋊D4 | Dic7.D4 | C28⋊Q8 | C28⋊1D4 | C4×C7⋊D4 | C28⋊2D4 | C28⋊D4 | D4×C28 | C2×C4○D28 | C4×D7 | C4×D4 | Dic7 | C28 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 3 | 4 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4^2._{228}D_{14} % in TeX
G:=Group("C4^2.228D14"); // GroupNames label
G:=SmallGroup(448,1001);
// by ID
G=gap.SmallGroup(448,1001);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,100,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀