metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.180D14, C14.412- (1+4), C14.862+ (1+4), C4⋊Q8⋊18D7, C4⋊C4.127D14, (C2×Q8).90D14, D14⋊3Q8⋊39C2, D14⋊Q8⋊49C2, C42⋊2D7⋊20C2, Dic7.Q8⋊43C2, Dic7⋊Q8⋊28C2, (C2×C14).279C24, (C4×C28).274C22, (C2×C28).641C23, D14⋊C4.76C22, D14.5D4.5C2, C28.23D4.9C2, C2.90(D4⋊6D14), (C2×D28).174C22, Dic7⋊C4.88C22, C4⋊Dic7.256C22, (Q8×C14).146C22, C22.300(C23×D7), C7⋊6(C22.57C24), (C4×Dic7).168C22, (C2×Dic7).147C23, (C22×D7).124C23, C2.42(Q8.10D14), (C2×Dic14).193C22, (C7×C4⋊Q8)⋊21C2, C4⋊C4⋊D7⋊48C2, (C2×C4×D7).152C22, (C7×C4⋊C4).222C22, (C2×C4).222(C22×D7), SmallGroup(448,1188)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 876 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4, Q8 [×3], C23 [×2], D7 [×2], C14, C14 [×2], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8, Dic7 [×6], C28 [×7], D14 [×6], C2×C14, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8, C4⋊Q8, Dic14, C4×D7 [×2], D28, C2×Dic7 [×6], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×2], C22×D7 [×2], C22.57C24, C4×Dic7 [×2], Dic7⋊C4 [×10], C4⋊Dic7 [×2], D14⋊C4 [×10], C4×C28, C7×C4⋊C4 [×4], C2×Dic14, C2×C4×D7 [×2], C2×D28, Q8×C14 [×2], C42⋊2D7 [×2], Dic7.Q8 [×2], D14.5D4 [×2], D14⋊Q8 [×2], C4⋊C4⋊D7 [×2], Dic7⋊Q8, D14⋊3Q8 [×2], C28.23D4, C7×C4⋊Q8, C42.180D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4), 2- (1+4) [×2], C22×D7 [×7], C22.57C24, C23×D7, D4⋊6D14, Q8.10D14 [×2], C42.180D14
Generators and relations
G = < a,b,c,d | a4=b4=1, c14=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b, dcd-1=c13 >
►On 224 points
Generators in S224
(1 84 15 70)(2 71 16 57)(3 58 17 72)(4 73 18 59)(5 60 19 74)(6 75 20 61)(7 62 21 76)(8 77 22 63)(9 64 23 78)(10 79 24 65)(11 66 25 80)(12 81 26 67)(13 68 27 82)(14 83 28 69)(29 160 43 146)(30 147 44 161)(31 162 45 148)(32 149 46 163)(33 164 47 150)(34 151 48 165)(35 166 49 152)(36 153 50 167)(37 168 51 154)(38 155 52 141)(39 142 53 156)(40 157 54 143)(41 144 55 158)(42 159 56 145)(85 215 99 201)(86 202 100 216)(87 217 101 203)(88 204 102 218)(89 219 103 205)(90 206 104 220)(91 221 105 207)(92 208 106 222)(93 223 107 209)(94 210 108 224)(95 197 109 211)(96 212 110 198)(97 199 111 213)(98 214 112 200)(113 174 127 188)(114 189 128 175)(115 176 129 190)(116 191 130 177)(117 178 131 192)(118 193 132 179)(119 180 133 194)(120 195 134 181)(121 182 135 196)(122 169 136 183)(123 184 137 170)(124 171 138 185)(125 186 139 172)(126 173 140 187)
(1 160 208 117)(2 118 209 161)(3 162 210 119)(4 120 211 163)(5 164 212 121)(6 122 213 165)(7 166 214 123)(8 124 215 167)(9 168 216 125)(10 126 217 141)(11 142 218 127)(12 128 219 143)(13 144 220 129)(14 130 221 145)(15 146 222 131)(16 132 223 147)(17 148 224 133)(18 134 197 149)(19 150 198 135)(20 136 199 151)(21 152 200 137)(22 138 201 153)(23 154 202 139)(24 140 203 155)(25 156 204 113)(26 114 205 157)(27 158 206 115)(28 116 207 159)(29 92 192 70)(30 71 193 93)(31 94 194 72)(32 73 195 95)(33 96 196 74)(34 75 169 97)(35 98 170 76)(36 77 171 99)(37 100 172 78)(38 79 173 101)(39 102 174 80)(40 81 175 103)(41 104 176 82)(42 83 177 105)(43 106 178 84)(44 57 179 107)(45 108 180 58)(46 59 181 109)(47 110 182 60)(48 61 183 111)(49 112 184 62)(50 63 185 85)(51 86 186 64)(52 65 187 87)(53 88 188 66)(54 67 189 89)(55 90 190 68)(56 69 191 91)
(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)
(1 184 15 170)(2 169 16 183)(3 182 17 196)(4 195 18 181)(5 180 19 194)(6 193 20 179)(7 178 21 192)(8 191 22 177)(9 176 23 190)(10 189 24 175)(11 174 25 188)(12 187 26 173)(13 172 27 186)(14 185 28 171)(29 214 43 200)(30 199 44 213)(31 212 45 198)(32 197 46 211)(33 210 47 224)(34 223 48 209)(35 208 49 222)(36 221 50 207)(37 206 51 220)(38 219 52 205)(39 204 53 218)(40 217 54 203)(41 202 55 216)(42 215 56 201)(57 151 71 165)(58 164 72 150)(59 149 73 163)(60 162 74 148)(61 147 75 161)(62 160 76 146)(63 145 77 159)(64 158 78 144)(65 143 79 157)(66 156 80 142)(67 141 81 155)(68 154 82 168)(69 167 83 153)(70 152 84 166)(85 130 99 116)(86 115 100 129)(87 128 101 114)(88 113 102 127)(89 126 103 140)(90 139 104 125)(91 124 105 138)(92 137 106 123)(93 122 107 136)(94 135 108 121)(95 120 109 134)(96 133 110 119)(97 118 111 132)(98 131 112 117)
G:=sub<Sym(224)| (1,84,15,70)(2,71,16,57)(3,58,17,72)(4,73,18,59)(5,60,19,74)(6,75,20,61)(7,62,21,76)(8,77,22,63)(9,64,23,78)(10,79,24,65)(11,66,25,80)(12,81,26,67)(13,68,27,82)(14,83,28,69)(29,160,43,146)(30,147,44,161)(31,162,45,148)(32,149,46,163)(33,164,47,150)(34,151,48,165)(35,166,49,152)(36,153,50,167)(37,168,51,154)(38,155,52,141)(39,142,53,156)(40,157,54,143)(41,144,55,158)(42,159,56,145)(85,215,99,201)(86,202,100,216)(87,217,101,203)(88,204,102,218)(89,219,103,205)(90,206,104,220)(91,221,105,207)(92,208,106,222)(93,223,107,209)(94,210,108,224)(95,197,109,211)(96,212,110,198)(97,199,111,213)(98,214,112,200)(113,174,127,188)(114,189,128,175)(115,176,129,190)(116,191,130,177)(117,178,131,192)(118,193,132,179)(119,180,133,194)(120,195,134,181)(121,182,135,196)(122,169,136,183)(123,184,137,170)(124,171,138,185)(125,186,139,172)(126,173,140,187), (1,160,208,117)(2,118,209,161)(3,162,210,119)(4,120,211,163)(5,164,212,121)(6,122,213,165)(7,166,214,123)(8,124,215,167)(9,168,216,125)(10,126,217,141)(11,142,218,127)(12,128,219,143)(13,144,220,129)(14,130,221,145)(15,146,222,131)(16,132,223,147)(17,148,224,133)(18,134,197,149)(19,150,198,135)(20,136,199,151)(21,152,200,137)(22,138,201,153)(23,154,202,139)(24,140,203,155)(25,156,204,113)(26,114,205,157)(27,158,206,115)(28,116,207,159)(29,92,192,70)(30,71,193,93)(31,94,194,72)(32,73,195,95)(33,96,196,74)(34,75,169,97)(35,98,170,76)(36,77,171,99)(37,100,172,78)(38,79,173,101)(39,102,174,80)(40,81,175,103)(41,104,176,82)(42,83,177,105)(43,106,178,84)(44,57,179,107)(45,108,180,58)(46,59,181,109)(47,110,182,60)(48,61,183,111)(49,112,184,62)(50,63,185,85)(51,86,186,64)(52,65,187,87)(53,88,188,66)(54,67,189,89)(55,90,190,68)(56,69,191,91), (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), (1,184,15,170)(2,169,16,183)(3,182,17,196)(4,195,18,181)(5,180,19,194)(6,193,20,179)(7,178,21,192)(8,191,22,177)(9,176,23,190)(10,189,24,175)(11,174,25,188)(12,187,26,173)(13,172,27,186)(14,185,28,171)(29,214,43,200)(30,199,44,213)(31,212,45,198)(32,197,46,211)(33,210,47,224)(34,223,48,209)(35,208,49,222)(36,221,50,207)(37,206,51,220)(38,219,52,205)(39,204,53,218)(40,217,54,203)(41,202,55,216)(42,215,56,201)(57,151,71,165)(58,164,72,150)(59,149,73,163)(60,162,74,148)(61,147,75,161)(62,160,76,146)(63,145,77,159)(64,158,78,144)(65,143,79,157)(66,156,80,142)(67,141,81,155)(68,154,82,168)(69,167,83,153)(70,152,84,166)(85,130,99,116)(86,115,100,129)(87,128,101,114)(88,113,102,127)(89,126,103,140)(90,139,104,125)(91,124,105,138)(92,137,106,123)(93,122,107,136)(94,135,108,121)(95,120,109,134)(96,133,110,119)(97,118,111,132)(98,131,112,117)>;
G:=Group( (1,84,15,70)(2,71,16,57)(3,58,17,72)(4,73,18,59)(5,60,19,74)(6,75,20,61)(7,62,21,76)(8,77,22,63)(9,64,23,78)(10,79,24,65)(11,66,25,80)(12,81,26,67)(13,68,27,82)(14,83,28,69)(29,160,43,146)(30,147,44,161)(31,162,45,148)(32,149,46,163)(33,164,47,150)(34,151,48,165)(35,166,49,152)(36,153,50,167)(37,168,51,154)(38,155,52,141)(39,142,53,156)(40,157,54,143)(41,144,55,158)(42,159,56,145)(85,215,99,201)(86,202,100,216)(87,217,101,203)(88,204,102,218)(89,219,103,205)(90,206,104,220)(91,221,105,207)(92,208,106,222)(93,223,107,209)(94,210,108,224)(95,197,109,211)(96,212,110,198)(97,199,111,213)(98,214,112,200)(113,174,127,188)(114,189,128,175)(115,176,129,190)(116,191,130,177)(117,178,131,192)(118,193,132,179)(119,180,133,194)(120,195,134,181)(121,182,135,196)(122,169,136,183)(123,184,137,170)(124,171,138,185)(125,186,139,172)(126,173,140,187), (1,160,208,117)(2,118,209,161)(3,162,210,119)(4,120,211,163)(5,164,212,121)(6,122,213,165)(7,166,214,123)(8,124,215,167)(9,168,216,125)(10,126,217,141)(11,142,218,127)(12,128,219,143)(13,144,220,129)(14,130,221,145)(15,146,222,131)(16,132,223,147)(17,148,224,133)(18,134,197,149)(19,150,198,135)(20,136,199,151)(21,152,200,137)(22,138,201,153)(23,154,202,139)(24,140,203,155)(25,156,204,113)(26,114,205,157)(27,158,206,115)(28,116,207,159)(29,92,192,70)(30,71,193,93)(31,94,194,72)(32,73,195,95)(33,96,196,74)(34,75,169,97)(35,98,170,76)(36,77,171,99)(37,100,172,78)(38,79,173,101)(39,102,174,80)(40,81,175,103)(41,104,176,82)(42,83,177,105)(43,106,178,84)(44,57,179,107)(45,108,180,58)(46,59,181,109)(47,110,182,60)(48,61,183,111)(49,112,184,62)(50,63,185,85)(51,86,186,64)(52,65,187,87)(53,88,188,66)(54,67,189,89)(55,90,190,68)(56,69,191,91), (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), (1,184,15,170)(2,169,16,183)(3,182,17,196)(4,195,18,181)(5,180,19,194)(6,193,20,179)(7,178,21,192)(8,191,22,177)(9,176,23,190)(10,189,24,175)(11,174,25,188)(12,187,26,173)(13,172,27,186)(14,185,28,171)(29,214,43,200)(30,199,44,213)(31,212,45,198)(32,197,46,211)(33,210,47,224)(34,223,48,209)(35,208,49,222)(36,221,50,207)(37,206,51,220)(38,219,52,205)(39,204,53,218)(40,217,54,203)(41,202,55,216)(42,215,56,201)(57,151,71,165)(58,164,72,150)(59,149,73,163)(60,162,74,148)(61,147,75,161)(62,160,76,146)(63,145,77,159)(64,158,78,144)(65,143,79,157)(66,156,80,142)(67,141,81,155)(68,154,82,168)(69,167,83,153)(70,152,84,166)(85,130,99,116)(86,115,100,129)(87,128,101,114)(88,113,102,127)(89,126,103,140)(90,139,104,125)(91,124,105,138)(92,137,106,123)(93,122,107,136)(94,135,108,121)(95,120,109,134)(96,133,110,119)(97,118,111,132)(98,131,112,117) );
G=PermutationGroup([(1,84,15,70),(2,71,16,57),(3,58,17,72),(4,73,18,59),(5,60,19,74),(6,75,20,61),(7,62,21,76),(8,77,22,63),(9,64,23,78),(10,79,24,65),(11,66,25,80),(12,81,26,67),(13,68,27,82),(14,83,28,69),(29,160,43,146),(30,147,44,161),(31,162,45,148),(32,149,46,163),(33,164,47,150),(34,151,48,165),(35,166,49,152),(36,153,50,167),(37,168,51,154),(38,155,52,141),(39,142,53,156),(40,157,54,143),(41,144,55,158),(42,159,56,145),(85,215,99,201),(86,202,100,216),(87,217,101,203),(88,204,102,218),(89,219,103,205),(90,206,104,220),(91,221,105,207),(92,208,106,222),(93,223,107,209),(94,210,108,224),(95,197,109,211),(96,212,110,198),(97,199,111,213),(98,214,112,200),(113,174,127,188),(114,189,128,175),(115,176,129,190),(116,191,130,177),(117,178,131,192),(118,193,132,179),(119,180,133,194),(120,195,134,181),(121,182,135,196),(122,169,136,183),(123,184,137,170),(124,171,138,185),(125,186,139,172),(126,173,140,187)], [(1,160,208,117),(2,118,209,161),(3,162,210,119),(4,120,211,163),(5,164,212,121),(6,122,213,165),(7,166,214,123),(8,124,215,167),(9,168,216,125),(10,126,217,141),(11,142,218,127),(12,128,219,143),(13,144,220,129),(14,130,221,145),(15,146,222,131),(16,132,223,147),(17,148,224,133),(18,134,197,149),(19,150,198,135),(20,136,199,151),(21,152,200,137),(22,138,201,153),(23,154,202,139),(24,140,203,155),(25,156,204,113),(26,114,205,157),(27,158,206,115),(28,116,207,159),(29,92,192,70),(30,71,193,93),(31,94,194,72),(32,73,195,95),(33,96,196,74),(34,75,169,97),(35,98,170,76),(36,77,171,99),(37,100,172,78),(38,79,173,101),(39,102,174,80),(40,81,175,103),(41,104,176,82),(42,83,177,105),(43,106,178,84),(44,57,179,107),(45,108,180,58),(46,59,181,109),(47,110,182,60),(48,61,183,111),(49,112,184,62),(50,63,185,85),(51,86,186,64),(52,65,187,87),(53,88,188,66),(54,67,189,89),(55,90,190,68),(56,69,191,91)], [(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)], [(1,184,15,170),(2,169,16,183),(3,182,17,196),(4,195,18,181),(5,180,19,194),(6,193,20,179),(7,178,21,192),(8,191,22,177),(9,176,23,190),(10,189,24,175),(11,174,25,188),(12,187,26,173),(13,172,27,186),(14,185,28,171),(29,214,43,200),(30,199,44,213),(31,212,45,198),(32,197,46,211),(33,210,47,224),(34,223,48,209),(35,208,49,222),(36,221,50,207),(37,206,51,220),(38,219,52,205),(39,204,53,218),(40,217,54,203),(41,202,55,216),(42,215,56,201),(57,151,71,165),(58,164,72,150),(59,149,73,163),(60,162,74,148),(61,147,75,161),(62,160,76,146),(63,145,77,159),(64,158,78,144),(65,143,79,157),(66,156,80,142),(67,141,81,155),(68,154,82,168),(69,167,83,153),(70,152,84,166),(85,130,99,116),(86,115,100,129),(87,128,101,114),(88,113,102,127),(89,126,103,140),(90,139,104,125),(91,124,105,138),(92,137,106,123),(93,122,107,136),(94,135,108,121),(95,120,109,134),(96,133,110,119),(97,118,111,132),(98,131,112,117)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 11 | 2 | 11 | 27 | 0 | 0 | 0 | 0 |
| 27 | 18 | 0 | 2 | 0 | 0 | 0 | 0 |
| 13 | 13 | 20 | 27 | 0 | 0 | 0 | 0 |
| 0 | 28 | 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 | 25 | 15 |
| 0 | 0 | 0 | 0 | 22 | 8 | 28 | 4 |
| 28 | 0 | 24 | 18 | 0 | 0 | 0 | 0 |
| 0 | 28 | 22 | 18 | 0 | 0 | 0 | 0 |
| 28 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 27 | 18 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 16 | 11 | 4 |
| 0 | 0 | 0 | 0 | 13 | 5 | 25 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 | 9 | 13 |
| 0 | 0 | 0 | 0 | 14 | 24 | 5 | 20 |
| 4 | 0 | 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 4 | 25 | 0 | 0 | 0 | 0 |
| 4 | 22 | 25 | 0 | 0 | 0 | 0 | 0 |
| 11 | 16 | 0 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 26 | 25 | 5 |
| 0 | 0 | 0 | 0 | 26 | 10 | 7 | 4 |
| 0 | 0 | 0 | 0 | 19 | 10 | 20 | 3 |
| 0 | 0 | 0 | 0 | 0 | 6 | 17 | 3 |
| 7 | 22 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 22 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 10 | 22 | 0 | 0 | 0 | 0 |
| 18 | 25 | 2 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 5 | 16 | 27 |
| 0 | 0 | 0 | 0 | 5 | 16 | 10 | 7 |
| 0 | 0 | 0 | 0 | 17 | 11 | 21 | 24 |
| 0 | 0 | 0 | 0 | 15 | 23 | 2 | 5 |
G:=sub<GL(8,GF(29))| [11,27,13,0,0,0,0,0,2,18,13,28,0,0,0,0,11,0,20,11,0,0,0,0,27,2,27,9,0,0,0,0,0,0,0,0,9,14,21,22,0,0,0,0,15,20,0,8,0,0,0,0,0,0,25,28,0,0,0,0,0,0,15,4],[28,0,28,27,0,0,0,0,0,28,1,18,0,0,0,0,24,22,1,0,0,0,0,0,18,18,0,1,0,0,0,0,0,0,0,0,24,13,0,14,0,0,0,0,16,5,15,24,0,0,0,0,11,25,9,5,0,0,0,0,4,0,13,20],[4,0,4,11,0,0,0,0,0,4,22,16,0,0,0,0,25,4,25,0,0,0,0,0,0,25,0,25,0,0,0,0,0,0,0,0,25,26,19,0,0,0,0,0,26,10,10,6,0,0,0,0,25,7,20,17,0,0,0,0,5,4,3,3],[7,3,4,18,0,0,0,0,22,22,0,25,0,0,0,0,0,0,10,2,0,0,0,0,0,0,22,19,0,0,0,0,0,0,0,0,16,5,17,15,0,0,0,0,5,16,11,23,0,0,0,0,16,10,21,2,0,0,0,0,27,7,24,5] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4G | 4H | ··· | 4M | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊6D14 | Q8.10D14 |
| kernel | C42.180D14 | C42⋊2D7 | Dic7.Q8 | D14.5D4 | D14⋊Q8 | C4⋊C4⋊D7 | Dic7⋊Q8 | D14⋊3Q8 | C28.23D4 | C7×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 3 | 3 | 12 | 6 | 1 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{180}D_{14} % in TeX
G:=Group("C4^2.180D14"); // GroupNames label
G:=SmallGroup(448,1188);
// by ID
G=gap.SmallGroup(448,1188);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,219,100,1571,570,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^13>;
// generators/relations
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