metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.140D14, C14.892- (1+4), C14.722+ (1+4), (C2×Q8).83D14, C4.4D4.9D7, (C2×D4).109D14, (C2×C28).78C23, C22⋊C4.34D14, C28.6Q8⋊28C2, Dic7⋊Q8⋊23C2, (C2×C14).216C24, (C4×C28).221C22, C4⋊Dic7.50C22, C2.74(D4⋊6D14), C23.38(C22×D7), C22⋊Dic14⋊39C2, (D4×C14).209C22, C23.D14⋊38C2, Dic7⋊C4.83C22, (C22×C14).46C23, (Q8×C14).125C22, C22.237(C23×D7), C23.D7.53C22, C7⋊3(C22.57C24), (C4×Dic7).132C22, (C2×Dic7).111C23, C23.18D14.6C2, C2.50(D4.10D14), (C2×Dic14).176C22, (C22×Dic7).141C22, (C7×C4.4D4).7C2, (C2×C4).192(C22×D7), (C7×C22⋊C4).63C22, SmallGroup(448,1125)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 780 in 196 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C7, C2×C4, C2×C4 [×4], C2×C4 [×10], D4, Q8 [×3], C23 [×2], C14, C14 [×2], C14 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×16], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic7 [×8], C28 [×5], C2×C14, C2×C14 [×6], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], Dic14 [×2], C2×Dic7 [×8], C2×Dic7 [×2], C2×C28, C2×C28 [×4], C7×D4, C7×Q8, C22×C14 [×2], C22.57C24, C4×Dic7 [×2], Dic7⋊C4 [×12], C4⋊Dic7 [×4], C23.D7 [×6], C4×C28, C7×C22⋊C4 [×4], C2×Dic14 [×2], C22×Dic7 [×2], D4×C14, Q8×C14, C28.6Q8 [×2], C22⋊Dic14 [×4], C23.D14 [×4], C23.18D14 [×2], Dic7⋊Q8 [×2], C7×C4.4D4, C42.140D14
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, D4.10D14 [×2], C42.140D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >
►On 224 points
Generators in S224
(1 57 26 89)(2 65 27 97)(3 59 28 91)(4 67 22 85)(5 61 23 93)(6 69 24 87)(7 63 25 95)(8 92 15 60)(9 86 16 68)(10 94 17 62)(11 88 18 70)(12 96 19 64)(13 90 20 58)(14 98 21 66)(29 146 55 138)(30 154 56 132)(31 148 50 140)(32 142 51 134)(33 150 52 128)(34 144 53 136)(35 152 54 130)(36 127 44 149)(37 135 45 143)(38 129 46 151)(39 137 47 145)(40 131 48 153)(41 139 49 147)(42 133 43 141)(71 105 121 112)(72 221 122 214)(73 107 123 100)(74 223 124 216)(75 109 125 102)(76 211 126 218)(77 111 113 104)(78 213 114 220)(79 99 115 106)(80 215 116 222)(81 101 117 108)(82 217 118 224)(83 103 119 110)(84 219 120 212)(155 190 203 183)(156 170 204 177)(157 192 205 185)(158 172 206 179)(159 194 207 187)(160 174 208 181)(161 196 209 189)(162 176 210 169)(163 184 197 191)(164 178 198 171)(165 186 199 193)(166 180 200 173)(167 188 201 195)(168 182 202 175)
(1 201 12 208)(2 209 13 202)(3 203 14 210)(4 197 8 204)(5 205 9 198)(6 199 10 206)(7 207 11 200)(15 156 22 163)(16 164 23 157)(17 158 24 165)(18 166 25 159)(19 160 26 167)(20 168 27 161)(21 162 28 155)(29 80 40 73)(30 74 41 81)(31 82 42 75)(32 76 36 83)(33 84 37 77)(34 78 38 71)(35 72 39 79)(43 125 50 118)(44 119 51 126)(45 113 52 120)(46 121 53 114)(47 115 54 122)(48 123 55 116)(49 117 56 124)(57 195 96 181)(58 182 97 196)(59 183 98 169)(60 170 85 184)(61 185 86 171)(62 172 87 186)(63 187 88 173)(64 174 89 188)(65 189 90 175)(66 176 91 190)(67 191 92 177)(68 178 93 192)(69 193 94 179)(70 180 95 194)(99 152 221 137)(100 138 222 153)(101 154 223 139)(102 140 224 141)(103 142 211 127)(104 128 212 143)(105 144 213 129)(106 130 214 145)(107 146 215 131)(108 132 216 147)(109 148 217 133)(110 134 218 149)(111 150 219 135)(112 136 220 151)
(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 39 12 35)(2 38 13 34)(3 37 14 33)(4 36 8 32)(5 42 9 31)(6 41 10 30)(7 40 11 29)(15 51 22 44)(16 50 23 43)(17 56 24 49)(18 55 25 48)(19 54 26 47)(20 53 27 46)(21 52 28 45)(57 145 96 130)(58 144 97 129)(59 143 98 128)(60 142 85 127)(61 141 86 140)(62 154 87 139)(63 153 88 138)(64 152 89 137)(65 151 90 136)(66 150 91 135)(67 149 92 134)(68 148 93 133)(69 147 94 132)(70 146 95 131)(71 161 78 168)(72 160 79 167)(73 159 80 166)(74 158 81 165)(75 157 82 164)(76 156 83 163)(77 155 84 162)(99 195 221 181)(100 194 222 180)(101 193 223 179)(102 192 224 178)(103 191 211 177)(104 190 212 176)(105 189 213 175)(106 188 214 174)(107 187 215 173)(108 186 216 172)(109 185 217 171)(110 184 218 170)(111 183 219 169)(112 196 220 182)(113 203 120 210)(114 202 121 209)(115 201 122 208)(116 200 123 207)(117 199 124 206)(118 198 125 205)(119 197 126 204)
G:=sub<Sym(224)| (1,57,26,89)(2,65,27,97)(3,59,28,91)(4,67,22,85)(5,61,23,93)(6,69,24,87)(7,63,25,95)(8,92,15,60)(9,86,16,68)(10,94,17,62)(11,88,18,70)(12,96,19,64)(13,90,20,58)(14,98,21,66)(29,146,55,138)(30,154,56,132)(31,148,50,140)(32,142,51,134)(33,150,52,128)(34,144,53,136)(35,152,54,130)(36,127,44,149)(37,135,45,143)(38,129,46,151)(39,137,47,145)(40,131,48,153)(41,139,49,147)(42,133,43,141)(71,105,121,112)(72,221,122,214)(73,107,123,100)(74,223,124,216)(75,109,125,102)(76,211,126,218)(77,111,113,104)(78,213,114,220)(79,99,115,106)(80,215,116,222)(81,101,117,108)(82,217,118,224)(83,103,119,110)(84,219,120,212)(155,190,203,183)(156,170,204,177)(157,192,205,185)(158,172,206,179)(159,194,207,187)(160,174,208,181)(161,196,209,189)(162,176,210,169)(163,184,197,191)(164,178,198,171)(165,186,199,193)(166,180,200,173)(167,188,201,195)(168,182,202,175), (1,201,12,208)(2,209,13,202)(3,203,14,210)(4,197,8,204)(5,205,9,198)(6,199,10,206)(7,207,11,200)(15,156,22,163)(16,164,23,157)(17,158,24,165)(18,166,25,159)(19,160,26,167)(20,168,27,161)(21,162,28,155)(29,80,40,73)(30,74,41,81)(31,82,42,75)(32,76,36,83)(33,84,37,77)(34,78,38,71)(35,72,39,79)(43,125,50,118)(44,119,51,126)(45,113,52,120)(46,121,53,114)(47,115,54,122)(48,123,55,116)(49,117,56,124)(57,195,96,181)(58,182,97,196)(59,183,98,169)(60,170,85,184)(61,185,86,171)(62,172,87,186)(63,187,88,173)(64,174,89,188)(65,189,90,175)(66,176,91,190)(67,191,92,177)(68,178,93,192)(69,193,94,179)(70,180,95,194)(99,152,221,137)(100,138,222,153)(101,154,223,139)(102,140,224,141)(103,142,211,127)(104,128,212,143)(105,144,213,129)(106,130,214,145)(107,146,215,131)(108,132,216,147)(109,148,217,133)(110,134,218,149)(111,150,219,135)(112,136,220,151), (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), (1,39,12,35)(2,38,13,34)(3,37,14,33)(4,36,8,32)(5,42,9,31)(6,41,10,30)(7,40,11,29)(15,51,22,44)(16,50,23,43)(17,56,24,49)(18,55,25,48)(19,54,26,47)(20,53,27,46)(21,52,28,45)(57,145,96,130)(58,144,97,129)(59,143,98,128)(60,142,85,127)(61,141,86,140)(62,154,87,139)(63,153,88,138)(64,152,89,137)(65,151,90,136)(66,150,91,135)(67,149,92,134)(68,148,93,133)(69,147,94,132)(70,146,95,131)(71,161,78,168)(72,160,79,167)(73,159,80,166)(74,158,81,165)(75,157,82,164)(76,156,83,163)(77,155,84,162)(99,195,221,181)(100,194,222,180)(101,193,223,179)(102,192,224,178)(103,191,211,177)(104,190,212,176)(105,189,213,175)(106,188,214,174)(107,187,215,173)(108,186,216,172)(109,185,217,171)(110,184,218,170)(111,183,219,169)(112,196,220,182)(113,203,120,210)(114,202,121,209)(115,201,122,208)(116,200,123,207)(117,199,124,206)(118,198,125,205)(119,197,126,204)>;
G:=Group( (1,57,26,89)(2,65,27,97)(3,59,28,91)(4,67,22,85)(5,61,23,93)(6,69,24,87)(7,63,25,95)(8,92,15,60)(9,86,16,68)(10,94,17,62)(11,88,18,70)(12,96,19,64)(13,90,20,58)(14,98,21,66)(29,146,55,138)(30,154,56,132)(31,148,50,140)(32,142,51,134)(33,150,52,128)(34,144,53,136)(35,152,54,130)(36,127,44,149)(37,135,45,143)(38,129,46,151)(39,137,47,145)(40,131,48,153)(41,139,49,147)(42,133,43,141)(71,105,121,112)(72,221,122,214)(73,107,123,100)(74,223,124,216)(75,109,125,102)(76,211,126,218)(77,111,113,104)(78,213,114,220)(79,99,115,106)(80,215,116,222)(81,101,117,108)(82,217,118,224)(83,103,119,110)(84,219,120,212)(155,190,203,183)(156,170,204,177)(157,192,205,185)(158,172,206,179)(159,194,207,187)(160,174,208,181)(161,196,209,189)(162,176,210,169)(163,184,197,191)(164,178,198,171)(165,186,199,193)(166,180,200,173)(167,188,201,195)(168,182,202,175), (1,201,12,208)(2,209,13,202)(3,203,14,210)(4,197,8,204)(5,205,9,198)(6,199,10,206)(7,207,11,200)(15,156,22,163)(16,164,23,157)(17,158,24,165)(18,166,25,159)(19,160,26,167)(20,168,27,161)(21,162,28,155)(29,80,40,73)(30,74,41,81)(31,82,42,75)(32,76,36,83)(33,84,37,77)(34,78,38,71)(35,72,39,79)(43,125,50,118)(44,119,51,126)(45,113,52,120)(46,121,53,114)(47,115,54,122)(48,123,55,116)(49,117,56,124)(57,195,96,181)(58,182,97,196)(59,183,98,169)(60,170,85,184)(61,185,86,171)(62,172,87,186)(63,187,88,173)(64,174,89,188)(65,189,90,175)(66,176,91,190)(67,191,92,177)(68,178,93,192)(69,193,94,179)(70,180,95,194)(99,152,221,137)(100,138,222,153)(101,154,223,139)(102,140,224,141)(103,142,211,127)(104,128,212,143)(105,144,213,129)(106,130,214,145)(107,146,215,131)(108,132,216,147)(109,148,217,133)(110,134,218,149)(111,150,219,135)(112,136,220,151), (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), (1,39,12,35)(2,38,13,34)(3,37,14,33)(4,36,8,32)(5,42,9,31)(6,41,10,30)(7,40,11,29)(15,51,22,44)(16,50,23,43)(17,56,24,49)(18,55,25,48)(19,54,26,47)(20,53,27,46)(21,52,28,45)(57,145,96,130)(58,144,97,129)(59,143,98,128)(60,142,85,127)(61,141,86,140)(62,154,87,139)(63,153,88,138)(64,152,89,137)(65,151,90,136)(66,150,91,135)(67,149,92,134)(68,148,93,133)(69,147,94,132)(70,146,95,131)(71,161,78,168)(72,160,79,167)(73,159,80,166)(74,158,81,165)(75,157,82,164)(76,156,83,163)(77,155,84,162)(99,195,221,181)(100,194,222,180)(101,193,223,179)(102,192,224,178)(103,191,211,177)(104,190,212,176)(105,189,213,175)(106,188,214,174)(107,187,215,173)(108,186,216,172)(109,185,217,171)(110,184,218,170)(111,183,219,169)(112,196,220,182)(113,203,120,210)(114,202,121,209)(115,201,122,208)(116,200,123,207)(117,199,124,206)(118,198,125,205)(119,197,126,204) );
G=PermutationGroup([(1,57,26,89),(2,65,27,97),(3,59,28,91),(4,67,22,85),(5,61,23,93),(6,69,24,87),(7,63,25,95),(8,92,15,60),(9,86,16,68),(10,94,17,62),(11,88,18,70),(12,96,19,64),(13,90,20,58),(14,98,21,66),(29,146,55,138),(30,154,56,132),(31,148,50,140),(32,142,51,134),(33,150,52,128),(34,144,53,136),(35,152,54,130),(36,127,44,149),(37,135,45,143),(38,129,46,151),(39,137,47,145),(40,131,48,153),(41,139,49,147),(42,133,43,141),(71,105,121,112),(72,221,122,214),(73,107,123,100),(74,223,124,216),(75,109,125,102),(76,211,126,218),(77,111,113,104),(78,213,114,220),(79,99,115,106),(80,215,116,222),(81,101,117,108),(82,217,118,224),(83,103,119,110),(84,219,120,212),(155,190,203,183),(156,170,204,177),(157,192,205,185),(158,172,206,179),(159,194,207,187),(160,174,208,181),(161,196,209,189),(162,176,210,169),(163,184,197,191),(164,178,198,171),(165,186,199,193),(166,180,200,173),(167,188,201,195),(168,182,202,175)], [(1,201,12,208),(2,209,13,202),(3,203,14,210),(4,197,8,204),(5,205,9,198),(6,199,10,206),(7,207,11,200),(15,156,22,163),(16,164,23,157),(17,158,24,165),(18,166,25,159),(19,160,26,167),(20,168,27,161),(21,162,28,155),(29,80,40,73),(30,74,41,81),(31,82,42,75),(32,76,36,83),(33,84,37,77),(34,78,38,71),(35,72,39,79),(43,125,50,118),(44,119,51,126),(45,113,52,120),(46,121,53,114),(47,115,54,122),(48,123,55,116),(49,117,56,124),(57,195,96,181),(58,182,97,196),(59,183,98,169),(60,170,85,184),(61,185,86,171),(62,172,87,186),(63,187,88,173),(64,174,89,188),(65,189,90,175),(66,176,91,190),(67,191,92,177),(68,178,93,192),(69,193,94,179),(70,180,95,194),(99,152,221,137),(100,138,222,153),(101,154,223,139),(102,140,224,141),(103,142,211,127),(104,128,212,143),(105,144,213,129),(106,130,214,145),(107,146,215,131),(108,132,216,147),(109,148,217,133),(110,134,218,149),(111,150,219,135),(112,136,220,151)], [(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)], [(1,39,12,35),(2,38,13,34),(3,37,14,33),(4,36,8,32),(5,42,9,31),(6,41,10,30),(7,40,11,29),(15,51,22,44),(16,50,23,43),(17,56,24,49),(18,55,25,48),(19,54,26,47),(20,53,27,46),(21,52,28,45),(57,145,96,130),(58,144,97,129),(59,143,98,128),(60,142,85,127),(61,141,86,140),(62,154,87,139),(63,153,88,138),(64,152,89,137),(65,151,90,136),(66,150,91,135),(67,149,92,134),(68,148,93,133),(69,147,94,132),(70,146,95,131),(71,161,78,168),(72,160,79,167),(73,159,80,166),(74,158,81,165),(75,157,82,164),(76,156,83,163),(77,155,84,162),(99,195,221,181),(100,194,222,180),(101,193,223,179),(102,192,224,178),(103,191,211,177),(104,190,212,176),(105,189,213,175),(106,188,214,174),(107,187,215,173),(108,186,216,172),(109,185,217,171),(110,184,218,170),(111,183,219,169),(112,196,220,182),(113,203,120,210),(114,202,121,209),(115,201,122,208),(116,200,123,207),(117,199,124,206),(118,198,125,205),(119,197,126,204)])
Matrix representation ►G ⊆ GL10(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 24 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 28 | 22 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 17 | 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 17 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 17 | 8 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 9 | 22 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 11 | 28 | 24 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 15 | 12 | 1 |
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 23 | 6 | 17 | 28 |
| 23 | 21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 6 | 11 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 0 | 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 20 | 8 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 21 | 23 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 21 | 12 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 21 | 14 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 19 | 15 | 2 |
G:=sub<GL(10,GF(29))| [28,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,1,13,12,11,0,0,0,0,0,0,0,28,0,17,0,0,0,0,0,0,24,22,28,16,0,0,0,0,0,0,0,5,0,1,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,2,17],[28,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,21,7,20,26,0,0,0,0,0,0,28,8,17,9,0,0,0,0,0,0,0,0,8,22,0,0,0,0,0,0,0,0,1,21,0,0,0,0,0,0,0,0,0,0,0,28,28,27,0,0,0,0,0,0,1,0,11,15,0,0,0,0,0,0,0,0,28,12,0,0,0,0,0,0,0,0,24,1],[19,10,0,0,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,23,0,0,0,0,0,0,0,28,28,6,0,0,0,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,0,28],[23,8,0,0,0,0,0,0,0,0,21,6,0,0,0,0,0,0,0,0,0,0,7,21,20,0,0,0,0,0,0,0,0,6,0,20,0,0,0,0,0,0,12,11,22,8,0,0,0,0,0,0,0,17,0,23,0,0,0,0,0,0,0,0,0,0,21,6,0,19,0,0,0,0,0,0,23,21,21,19,0,0,0,0,0,0,12,12,14,15,0,0,0,0,0,0,0,2,1,2] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4E | 4F | ··· | 4M | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊6D14 | D4.10D14 |
| kernel | C42.140D14 | C28.6Q8 | C22⋊Dic14 | C23.D14 | C23.18D14 | Dic7⋊Q8 | C7×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 2 | 2 | 1 | 3 | 3 | 12 | 3 | 3 | 1 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{140}D_{14} % in TeX
G:=Group("C4^2.140D14"); // GroupNames label
G:=SmallGroup(448,1125);
// by ID
G=gap.SmallGroup(448,1125);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,219,184,1571,570,297,136,18822]);
// 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=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀