metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊5D28, C42.128D14, C14.112- (1+4), (C4×Q8)⋊9D7, (C7×Q8)⋊10D4, (C4×D28)⋊38C2, (Q8×C28)⋊11C2, C7⋊2(Q8⋊5D4), C4.25(C2×D28), C28.57(C2×D4), C28⋊1D4⋊17C2, C4⋊C4.295D14, D14⋊13(C4○D4), D14⋊2Q8⋊18C2, C4.D28⋊20C2, D14⋊C4.6C22, (C2×Q8).204D14, C14.19(C22×D4), C2.21(C22×D28), (C2×C14).120C24, (C4×C28).172C22, (C2×C28).498C23, (C2×D28).216C22, C4⋊Dic7.306C22, (Q8×C14).220C22, (C2×Dic7).54C23, (C22×D7).45C23, C22.141(C23×D7), C2.12(Q8.10D14), (C2×Dic14).149C22, (C2×Q8×D7)⋊4C2, C2.29(D7×C4○D4), (C2×Q8⋊2D7)⋊3C2, (C2×C4×D7).72C22, C14.145(C2×C4○D4), (C7×C4⋊C4).348C22, (C2×C4).168(C22×D7), SmallGroup(448,1029)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1476 in 290 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×8], C22, C22 [×13], C7, C2×C4, C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×4], Q8 [×6], C23 [×4], D7 [×5], C14 [×3], C42 [×3], C22⋊C4 [×10], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×6], C2×D4 [×6], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic7 [×4], C28 [×6], C28 [×4], D14 [×2], D14 [×11], C2×C14, C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic14 [×6], C4×D7 [×12], D28 [×12], C2×Dic7, C2×Dic7 [×3], C2×C28, C2×C28 [×6], C7×Q8 [×4], C22×D7, C22×D7 [×3], Q8⋊5D4, C4⋊Dic7 [×3], D14⋊C4, D14⋊C4 [×9], C4×C28 [×3], C7×C4⋊C4 [×3], C2×Dic14 [×3], C2×C4×D7 [×6], C2×D28 [×6], Q8×D7 [×4], Q8⋊2D7 [×4], Q8×C14, C4×D28 [×3], C4.D28 [×3], C28⋊1D4 [×3], D14⋊2Q8 [×3], Q8×C28, C2×Q8×D7, C2×Q8⋊2D7, Q8⋊5D28
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2- (1+4), D28 [×4], C22×D7 [×7], Q8⋊5D4, C2×D28 [×6], C23×D7, C22×D28, Q8.10D14, D7×C4○D4, Q8⋊5D28
Generators and relations
G = < a,b,c,d | a4=c28=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 224 points
Generators in S224
(1 212 71 41)(2 213 72 42)(3 214 73 43)(4 215 74 44)(5 216 75 45)(6 217 76 46)(7 218 77 47)(8 219 78 48)(9 220 79 49)(10 221 80 50)(11 222 81 51)(12 223 82 52)(13 224 83 53)(14 197 84 54)(15 198 57 55)(16 199 58 56)(17 200 59 29)(18 201 60 30)(19 202 61 31)(20 203 62 32)(21 204 63 33)(22 205 64 34)(23 206 65 35)(24 207 66 36)(25 208 67 37)(26 209 68 38)(27 210 69 39)(28 211 70 40)(85 131 192 166)(86 132 193 167)(87 133 194 168)(88 134 195 141)(89 135 196 142)(90 136 169 143)(91 137 170 144)(92 138 171 145)(93 139 172 146)(94 140 173 147)(95 113 174 148)(96 114 175 149)(97 115 176 150)(98 116 177 151)(99 117 178 152)(100 118 179 153)(101 119 180 154)(102 120 181 155)(103 121 182 156)(104 122 183 157)(105 123 184 158)(106 124 185 159)(107 125 186 160)(108 126 187 161)(109 127 188 162)(110 128 189 163)(111 129 190 164)(112 130 191 165)
(1 86 71 193)(2 194 72 87)(3 88 73 195)(4 196 74 89)(5 90 75 169)(6 170 76 91)(7 92 77 171)(8 172 78 93)(9 94 79 173)(10 174 80 95)(11 96 81 175)(12 176 82 97)(13 98 83 177)(14 178 84 99)(15 100 57 179)(16 180 58 101)(17 102 59 181)(18 182 60 103)(19 104 61 183)(20 184 62 105)(21 106 63 185)(22 186 64 107)(23 108 65 187)(24 188 66 109)(25 110 67 189)(26 190 68 111)(27 112 69 191)(28 192 70 85)(29 120 200 155)(30 156 201 121)(31 122 202 157)(32 158 203 123)(33 124 204 159)(34 160 205 125)(35 126 206 161)(36 162 207 127)(37 128 208 163)(38 164 209 129)(39 130 210 165)(40 166 211 131)(41 132 212 167)(42 168 213 133)(43 134 214 141)(44 142 215 135)(45 136 216 143)(46 144 217 137)(47 138 218 145)(48 146 219 139)(49 140 220 147)(50 148 221 113)(51 114 222 149)(52 150 223 115)(53 116 224 151)(54 152 197 117)(55 118 198 153)(56 154 199 119)
(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 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 60)(12 59)(13 58)(14 57)(15 84)(16 83)(17 82)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 223)(30 222)(31 221)(32 220)(33 219)(34 218)(35 217)(36 216)(37 215)(38 214)(39 213)(40 212)(41 211)(42 210)(43 209)(44 208)(45 207)(46 206)(47 205)(48 204)(49 203)(50 202)(51 201)(52 200)(53 199)(54 198)(55 197)(56 224)(85 193)(86 192)(87 191)(88 190)(89 189)(90 188)(91 187)(92 186)(93 185)(94 184)(95 183)(96 182)(97 181)(98 180)(99 179)(100 178)(101 177)(102 176)(103 175)(104 174)(105 173)(106 172)(107 171)(108 170)(109 169)(110 196)(111 195)(112 194)(113 157)(114 156)(115 155)(116 154)(117 153)(118 152)(119 151)(120 150)(121 149)(122 148)(123 147)(124 146)(125 145)(126 144)(127 143)(128 142)(129 141)(130 168)(131 167)(132 166)(133 165)(134 164)(135 163)(136 162)(137 161)(138 160)(139 159)(140 158)
G:=sub<Sym(224)| (1,212,71,41)(2,213,72,42)(3,214,73,43)(4,215,74,44)(5,216,75,45)(6,217,76,46)(7,218,77,47)(8,219,78,48)(9,220,79,49)(10,221,80,50)(11,222,81,51)(12,223,82,52)(13,224,83,53)(14,197,84,54)(15,198,57,55)(16,199,58,56)(17,200,59,29)(18,201,60,30)(19,202,61,31)(20,203,62,32)(21,204,63,33)(22,205,64,34)(23,206,65,35)(24,207,66,36)(25,208,67,37)(26,209,68,38)(27,210,69,39)(28,211,70,40)(85,131,192,166)(86,132,193,167)(87,133,194,168)(88,134,195,141)(89,135,196,142)(90,136,169,143)(91,137,170,144)(92,138,171,145)(93,139,172,146)(94,140,173,147)(95,113,174,148)(96,114,175,149)(97,115,176,150)(98,116,177,151)(99,117,178,152)(100,118,179,153)(101,119,180,154)(102,120,181,155)(103,121,182,156)(104,122,183,157)(105,123,184,158)(106,124,185,159)(107,125,186,160)(108,126,187,161)(109,127,188,162)(110,128,189,163)(111,129,190,164)(112,130,191,165), (1,86,71,193)(2,194,72,87)(3,88,73,195)(4,196,74,89)(5,90,75,169)(6,170,76,91)(7,92,77,171)(8,172,78,93)(9,94,79,173)(10,174,80,95)(11,96,81,175)(12,176,82,97)(13,98,83,177)(14,178,84,99)(15,100,57,179)(16,180,58,101)(17,102,59,181)(18,182,60,103)(19,104,61,183)(20,184,62,105)(21,106,63,185)(22,186,64,107)(23,108,65,187)(24,188,66,109)(25,110,67,189)(26,190,68,111)(27,112,69,191)(28,192,70,85)(29,120,200,155)(30,156,201,121)(31,122,202,157)(32,158,203,123)(33,124,204,159)(34,160,205,125)(35,126,206,161)(36,162,207,127)(37,128,208,163)(38,164,209,129)(39,130,210,165)(40,166,211,131)(41,132,212,167)(42,168,213,133)(43,134,214,141)(44,142,215,135)(45,136,216,143)(46,144,217,137)(47,138,218,145)(48,146,219,139)(49,140,220,147)(50,148,221,113)(51,114,222,149)(52,150,223,115)(53,116,224,151)(54,152,197,117)(55,118,198,153)(56,154,199,119), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,223)(30,222)(31,221)(32,220)(33,219)(34,218)(35,217)(36,216)(37,215)(38,214)(39,213)(40,212)(41,211)(42,210)(43,209)(44,208)(45,207)(46,206)(47,205)(48,204)(49,203)(50,202)(51,201)(52,200)(53,199)(54,198)(55,197)(56,224)(85,193)(86,192)(87,191)(88,190)(89,189)(90,188)(91,187)(92,186)(93,185)(94,184)(95,183)(96,182)(97,181)(98,180)(99,179)(100,178)(101,177)(102,176)(103,175)(104,174)(105,173)(106,172)(107,171)(108,170)(109,169)(110,196)(111,195)(112,194)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,143)(128,142)(129,141)(130,168)(131,167)(132,166)(133,165)(134,164)(135,163)(136,162)(137,161)(138,160)(139,159)(140,158)>;
G:=Group( (1,212,71,41)(2,213,72,42)(3,214,73,43)(4,215,74,44)(5,216,75,45)(6,217,76,46)(7,218,77,47)(8,219,78,48)(9,220,79,49)(10,221,80,50)(11,222,81,51)(12,223,82,52)(13,224,83,53)(14,197,84,54)(15,198,57,55)(16,199,58,56)(17,200,59,29)(18,201,60,30)(19,202,61,31)(20,203,62,32)(21,204,63,33)(22,205,64,34)(23,206,65,35)(24,207,66,36)(25,208,67,37)(26,209,68,38)(27,210,69,39)(28,211,70,40)(85,131,192,166)(86,132,193,167)(87,133,194,168)(88,134,195,141)(89,135,196,142)(90,136,169,143)(91,137,170,144)(92,138,171,145)(93,139,172,146)(94,140,173,147)(95,113,174,148)(96,114,175,149)(97,115,176,150)(98,116,177,151)(99,117,178,152)(100,118,179,153)(101,119,180,154)(102,120,181,155)(103,121,182,156)(104,122,183,157)(105,123,184,158)(106,124,185,159)(107,125,186,160)(108,126,187,161)(109,127,188,162)(110,128,189,163)(111,129,190,164)(112,130,191,165), (1,86,71,193)(2,194,72,87)(3,88,73,195)(4,196,74,89)(5,90,75,169)(6,170,76,91)(7,92,77,171)(8,172,78,93)(9,94,79,173)(10,174,80,95)(11,96,81,175)(12,176,82,97)(13,98,83,177)(14,178,84,99)(15,100,57,179)(16,180,58,101)(17,102,59,181)(18,182,60,103)(19,104,61,183)(20,184,62,105)(21,106,63,185)(22,186,64,107)(23,108,65,187)(24,188,66,109)(25,110,67,189)(26,190,68,111)(27,112,69,191)(28,192,70,85)(29,120,200,155)(30,156,201,121)(31,122,202,157)(32,158,203,123)(33,124,204,159)(34,160,205,125)(35,126,206,161)(36,162,207,127)(37,128,208,163)(38,164,209,129)(39,130,210,165)(40,166,211,131)(41,132,212,167)(42,168,213,133)(43,134,214,141)(44,142,215,135)(45,136,216,143)(46,144,217,137)(47,138,218,145)(48,146,219,139)(49,140,220,147)(50,148,221,113)(51,114,222,149)(52,150,223,115)(53,116,224,151)(54,152,197,117)(55,118,198,153)(56,154,199,119), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,223)(30,222)(31,221)(32,220)(33,219)(34,218)(35,217)(36,216)(37,215)(38,214)(39,213)(40,212)(41,211)(42,210)(43,209)(44,208)(45,207)(46,206)(47,205)(48,204)(49,203)(50,202)(51,201)(52,200)(53,199)(54,198)(55,197)(56,224)(85,193)(86,192)(87,191)(88,190)(89,189)(90,188)(91,187)(92,186)(93,185)(94,184)(95,183)(96,182)(97,181)(98,180)(99,179)(100,178)(101,177)(102,176)(103,175)(104,174)(105,173)(106,172)(107,171)(108,170)(109,169)(110,196)(111,195)(112,194)(113,157)(114,156)(115,155)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,143)(128,142)(129,141)(130,168)(131,167)(132,166)(133,165)(134,164)(135,163)(136,162)(137,161)(138,160)(139,159)(140,158) );
G=PermutationGroup([(1,212,71,41),(2,213,72,42),(3,214,73,43),(4,215,74,44),(5,216,75,45),(6,217,76,46),(7,218,77,47),(8,219,78,48),(9,220,79,49),(10,221,80,50),(11,222,81,51),(12,223,82,52),(13,224,83,53),(14,197,84,54),(15,198,57,55),(16,199,58,56),(17,200,59,29),(18,201,60,30),(19,202,61,31),(20,203,62,32),(21,204,63,33),(22,205,64,34),(23,206,65,35),(24,207,66,36),(25,208,67,37),(26,209,68,38),(27,210,69,39),(28,211,70,40),(85,131,192,166),(86,132,193,167),(87,133,194,168),(88,134,195,141),(89,135,196,142),(90,136,169,143),(91,137,170,144),(92,138,171,145),(93,139,172,146),(94,140,173,147),(95,113,174,148),(96,114,175,149),(97,115,176,150),(98,116,177,151),(99,117,178,152),(100,118,179,153),(101,119,180,154),(102,120,181,155),(103,121,182,156),(104,122,183,157),(105,123,184,158),(106,124,185,159),(107,125,186,160),(108,126,187,161),(109,127,188,162),(110,128,189,163),(111,129,190,164),(112,130,191,165)], [(1,86,71,193),(2,194,72,87),(3,88,73,195),(4,196,74,89),(5,90,75,169),(6,170,76,91),(7,92,77,171),(8,172,78,93),(9,94,79,173),(10,174,80,95),(11,96,81,175),(12,176,82,97),(13,98,83,177),(14,178,84,99),(15,100,57,179),(16,180,58,101),(17,102,59,181),(18,182,60,103),(19,104,61,183),(20,184,62,105),(21,106,63,185),(22,186,64,107),(23,108,65,187),(24,188,66,109),(25,110,67,189),(26,190,68,111),(27,112,69,191),(28,192,70,85),(29,120,200,155),(30,156,201,121),(31,122,202,157),(32,158,203,123),(33,124,204,159),(34,160,205,125),(35,126,206,161),(36,162,207,127),(37,128,208,163),(38,164,209,129),(39,130,210,165),(40,166,211,131),(41,132,212,167),(42,168,213,133),(43,134,214,141),(44,142,215,135),(45,136,216,143),(46,144,217,137),(47,138,218,145),(48,146,219,139),(49,140,220,147),(50,148,221,113),(51,114,222,149),(52,150,223,115),(53,116,224,151),(54,152,197,117),(55,118,198,153),(56,154,199,119)], [(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)], [(1,70),(2,69),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,62),(10,61),(11,60),(12,59),(13,58),(14,57),(15,84),(16,83),(17,82),(18,81),(19,80),(20,79),(21,78),(22,77),(23,76),(24,75),(25,74),(26,73),(27,72),(28,71),(29,223),(30,222),(31,221),(32,220),(33,219),(34,218),(35,217),(36,216),(37,215),(38,214),(39,213),(40,212),(41,211),(42,210),(43,209),(44,208),(45,207),(46,206),(47,205),(48,204),(49,203),(50,202),(51,201),(52,200),(53,199),(54,198),(55,197),(56,224),(85,193),(86,192),(87,191),(88,190),(89,189),(90,188),(91,187),(92,186),(93,185),(94,184),(95,183),(96,182),(97,181),(98,180),(99,179),(100,178),(101,177),(102,176),(103,175),(104,174),(105,173),(106,172),(107,171),(108,170),(109,169),(110,196),(111,195),(112,194),(113,157),(114,156),(115,155),(116,154),(117,153),(118,152),(119,151),(120,150),(121,149),(122,148),(123,147),(124,146),(125,145),(126,144),(127,143),(128,142),(129,141),(130,168),(131,167),(132,166),(133,165),(134,164),(135,163),(136,162),(137,161),(138,160),(139,159),(140,158)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 0 | 1 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 22 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 10 | 11 | 0 | 0 | 0 | 0 |
| 12 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 24 |
| 0 | 0 | 0 | 0 | 17 | 12 |
| 19 | 1 | 0 | 0 | 0 | 0 |
| 17 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,1,0,0,0,0,27,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,9,0,0,0,0,22,2],[10,12,0,0,0,0,11,22,0,0,0,0,0,0,0,1,0,0,0,0,28,0,0,0,0,0,0,0,17,17,0,0,0,0,24,12],[19,17,0,0,0,0,1,10,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,12,12,0,0,0,0,5,17] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D28 | 2- (1+4) | Q8.10D14 | D7×C4○D4 |
| kernel | Q8⋊5D28 | C4×D28 | C4.D28 | C28⋊1D4 | D14⋊2Q8 | Q8×C28 | C2×Q8×D7 | C2×Q8⋊2D7 | C7×Q8 | C4×Q8 | D14 | C42 | C4⋊C4 | C2×Q8 | Q8 | C14 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_5D_{28} % in TeX
G:=Group("Q8:5D28"); // GroupNames label
G:=SmallGroup(448,1029);
// by ID
G=gap.SmallGroup(448,1029);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,184,675,192,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^28=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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ℤ
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