metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.137D14, C14.702+ 1+4, C14.872- 1+4, C4.4D4⋊6D7, C42⋊2D7⋊7C2, (C2×Q8).81D14, D14⋊3Q8⋊27C2, (C4×Dic14)⋊43C2, (C2×D4).107D14, C22⋊C4.71D14, Dic7⋊Q8⋊20C2, Dic7⋊4D4⋊28C2, D14.D4⋊40C2, (C2×C28).629C23, (C4×C28).183C22, (C2×C14).213C24, D14⋊C4.59C22, Dic7⋊D4.5C2, C2.72(D4⋊6D14), C23.35(C22×D7), Dic7.11(C4○D4), C22⋊Dic14⋊37C2, Dic7.D4⋊37C2, (D4×C14).207C22, C23.D14⋊36C2, Dic7⋊C4.82C22, C4⋊Dic7.232C22, (C22×C14).43C23, (Q8×C14).122C22, (C22×D7).93C23, C22.234(C23×D7), C23.D7.50C22, C23.18D14⋊24C2, C23.11D14⋊16C2, C7⋊8(C22.36C24), (C2×Dic7).250C23, (C4×Dic7).213C22, C2.48(D4.10D14), (C2×Dic14).174C22, (C22×Dic7).138C22, C2.72(D7×C4○D4), (C7×C4.4D4)⋊7C2, C14.184(C2×C4○D4), (C2×C4×D7).119C22, (C2×C4).191(C22×D7), (C2×C7⋊D4).56C22, (C7×C22⋊C4).60C22, SmallGroup(448,1122)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.137D14
G = < a,b,c,d | a4=b4=c14=1, d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 940 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, C22×C14, C22.36C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C2×Dic14, C2×C4×D7, C22×Dic7, C2×C7⋊D4, D4×C14, Q8×C14, C4×Dic14, C42⋊2D7, C23.11D14, C22⋊Dic14, C23.D14, Dic7⋊4D4, D14.D4, Dic7.D4, C23.18D14, Dic7⋊D4, Dic7⋊Q8, D14⋊3Q8, C7×C4.4D4, C42.137D14
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D7, C22.36C24, C23×D7, D4⋊6D14, D7×C4○D4, D4.10D14, C42.137D14
►On 224 points
Generators in S224
(1 206 19 63)(2 200 20 57)(3 208 21 65)(4 202 15 59)(5 210 16 67)(6 204 17 61)(7 198 18 69)(8 209 43 66)(9 203 44 60)(10 197 45 68)(11 205 46 62)(12 199 47 70)(13 207 48 64)(14 201 49 58)(22 164 53 85)(23 158 54 93)(24 166 55 87)(25 160 56 95)(26 168 50 89)(27 162 51 97)(28 156 52 91)(29 96 39 161)(30 90 40 155)(31 98 41 163)(32 92 42 157)(33 86 36 165)(34 94 37 159)(35 88 38 167)(71 129 146 136)(72 181 147 174)(73 131 148 138)(74 169 149 176)(75 133 150 140)(76 171 151 178)(77 135 152 128)(78 173 153 180)(79 137 154 130)(80 175 141 182)(81 139 142 132)(82 177 143 170)(83 127 144 134)(84 179 145 172)(99 119 106 220)(100 214 107 113)(101 121 108 222)(102 216 109 115)(103 123 110 224)(104 218 111 117)(105 125 112 212)(114 189 215 196)(116 191 217 184)(118 193 219 186)(120 195 221 188)(122 183 223 190)(124 185 211 192)(126 187 213 194)
(1 137 12 174)(2 131 13 182)(3 139 14 176)(4 133 8 170)(5 127 9 178)(6 135 10 172)(7 129 11 180)(15 140 43 177)(16 134 44 171)(17 128 45 179)(18 136 46 173)(19 130 47 181)(20 138 48 175)(21 132 49 169)(22 191 42 103)(23 185 36 111)(24 193 37 105)(25 187 38 99)(26 195 39 107)(27 189 40 101)(28 183 41 109)(29 100 50 188)(30 108 51 196)(31 102 52 190)(32 110 53 184)(33 104 54 192)(34 112 55 186)(35 106 56 194)(57 73 64 141)(58 149 65 81)(59 75 66 143)(60 151 67 83)(61 77 68 145)(62 153 69 71)(63 79 70 147)(72 206 154 199)(74 208 142 201)(76 210 144 203)(78 198 146 205)(80 200 148 207)(82 202 150 209)(84 204 152 197)(85 116 92 224)(86 218 93 124)(87 118 94 212)(88 220 95 126)(89 120 96 214)(90 222 97 114)(91 122 98 216)(113 168 221 161)(115 156 223 163)(117 158 211 165)(119 160 213 167)(121 162 215 155)(123 164 217 157)(125 166 219 159)
(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 42 12 22)(2 41 13 28)(3 40 14 27)(4 39 8 26)(5 38 9 25)(6 37 10 24)(7 36 11 23)(15 29 43 50)(16 35 44 56)(17 34 45 55)(18 33 46 54)(19 32 47 53)(20 31 48 52)(21 30 49 51)(57 91 64 98)(58 90 65 97)(59 89 66 96)(60 88 67 95)(61 87 68 94)(62 86 69 93)(63 85 70 92)(71 117 153 211)(72 116 154 224)(73 115 141 223)(74 114 142 222)(75 113 143 221)(76 126 144 220)(77 125 145 219)(78 124 146 218)(79 123 147 217)(80 122 148 216)(81 121 149 215)(82 120 150 214)(83 119 151 213)(84 118 152 212)(99 134 187 171)(100 133 188 170)(101 132 189 169)(102 131 190 182)(103 130 191 181)(104 129 192 180)(105 128 193 179)(106 127 194 178)(107 140 195 177)(108 139 196 176)(109 138 183 175)(110 137 184 174)(111 136 185 173)(112 135 186 172)(155 208 162 201)(156 207 163 200)(157 206 164 199)(158 205 165 198)(159 204 166 197)(160 203 167 210)(161 202 168 209)
G:=sub<Sym(224)| (1,206,19,63)(2,200,20,57)(3,208,21,65)(4,202,15,59)(5,210,16,67)(6,204,17,61)(7,198,18,69)(8,209,43,66)(9,203,44,60)(10,197,45,68)(11,205,46,62)(12,199,47,70)(13,207,48,64)(14,201,49,58)(22,164,53,85)(23,158,54,93)(24,166,55,87)(25,160,56,95)(26,168,50,89)(27,162,51,97)(28,156,52,91)(29,96,39,161)(30,90,40,155)(31,98,41,163)(32,92,42,157)(33,86,36,165)(34,94,37,159)(35,88,38,167)(71,129,146,136)(72,181,147,174)(73,131,148,138)(74,169,149,176)(75,133,150,140)(76,171,151,178)(77,135,152,128)(78,173,153,180)(79,137,154,130)(80,175,141,182)(81,139,142,132)(82,177,143,170)(83,127,144,134)(84,179,145,172)(99,119,106,220)(100,214,107,113)(101,121,108,222)(102,216,109,115)(103,123,110,224)(104,218,111,117)(105,125,112,212)(114,189,215,196)(116,191,217,184)(118,193,219,186)(120,195,221,188)(122,183,223,190)(124,185,211,192)(126,187,213,194), (1,137,12,174)(2,131,13,182)(3,139,14,176)(4,133,8,170)(5,127,9,178)(6,135,10,172)(7,129,11,180)(15,140,43,177)(16,134,44,171)(17,128,45,179)(18,136,46,173)(19,130,47,181)(20,138,48,175)(21,132,49,169)(22,191,42,103)(23,185,36,111)(24,193,37,105)(25,187,38,99)(26,195,39,107)(27,189,40,101)(28,183,41,109)(29,100,50,188)(30,108,51,196)(31,102,52,190)(32,110,53,184)(33,104,54,192)(34,112,55,186)(35,106,56,194)(57,73,64,141)(58,149,65,81)(59,75,66,143)(60,151,67,83)(61,77,68,145)(62,153,69,71)(63,79,70,147)(72,206,154,199)(74,208,142,201)(76,210,144,203)(78,198,146,205)(80,200,148,207)(82,202,150,209)(84,204,152,197)(85,116,92,224)(86,218,93,124)(87,118,94,212)(88,220,95,126)(89,120,96,214)(90,222,97,114)(91,122,98,216)(113,168,221,161)(115,156,223,163)(117,158,211,165)(119,160,213,167)(121,162,215,155)(123,164,217,157)(125,166,219,159), (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), (1,42,12,22)(2,41,13,28)(3,40,14,27)(4,39,8,26)(5,38,9,25)(6,37,10,24)(7,36,11,23)(15,29,43,50)(16,35,44,56)(17,34,45,55)(18,33,46,54)(19,32,47,53)(20,31,48,52)(21,30,49,51)(57,91,64,98)(58,90,65,97)(59,89,66,96)(60,88,67,95)(61,87,68,94)(62,86,69,93)(63,85,70,92)(71,117,153,211)(72,116,154,224)(73,115,141,223)(74,114,142,222)(75,113,143,221)(76,126,144,220)(77,125,145,219)(78,124,146,218)(79,123,147,217)(80,122,148,216)(81,121,149,215)(82,120,150,214)(83,119,151,213)(84,118,152,212)(99,134,187,171)(100,133,188,170)(101,132,189,169)(102,131,190,182)(103,130,191,181)(104,129,192,180)(105,128,193,179)(106,127,194,178)(107,140,195,177)(108,139,196,176)(109,138,183,175)(110,137,184,174)(111,136,185,173)(112,135,186,172)(155,208,162,201)(156,207,163,200)(157,206,164,199)(158,205,165,198)(159,204,166,197)(160,203,167,210)(161,202,168,209)>;
G:=Group( (1,206,19,63)(2,200,20,57)(3,208,21,65)(4,202,15,59)(5,210,16,67)(6,204,17,61)(7,198,18,69)(8,209,43,66)(9,203,44,60)(10,197,45,68)(11,205,46,62)(12,199,47,70)(13,207,48,64)(14,201,49,58)(22,164,53,85)(23,158,54,93)(24,166,55,87)(25,160,56,95)(26,168,50,89)(27,162,51,97)(28,156,52,91)(29,96,39,161)(30,90,40,155)(31,98,41,163)(32,92,42,157)(33,86,36,165)(34,94,37,159)(35,88,38,167)(71,129,146,136)(72,181,147,174)(73,131,148,138)(74,169,149,176)(75,133,150,140)(76,171,151,178)(77,135,152,128)(78,173,153,180)(79,137,154,130)(80,175,141,182)(81,139,142,132)(82,177,143,170)(83,127,144,134)(84,179,145,172)(99,119,106,220)(100,214,107,113)(101,121,108,222)(102,216,109,115)(103,123,110,224)(104,218,111,117)(105,125,112,212)(114,189,215,196)(116,191,217,184)(118,193,219,186)(120,195,221,188)(122,183,223,190)(124,185,211,192)(126,187,213,194), (1,137,12,174)(2,131,13,182)(3,139,14,176)(4,133,8,170)(5,127,9,178)(6,135,10,172)(7,129,11,180)(15,140,43,177)(16,134,44,171)(17,128,45,179)(18,136,46,173)(19,130,47,181)(20,138,48,175)(21,132,49,169)(22,191,42,103)(23,185,36,111)(24,193,37,105)(25,187,38,99)(26,195,39,107)(27,189,40,101)(28,183,41,109)(29,100,50,188)(30,108,51,196)(31,102,52,190)(32,110,53,184)(33,104,54,192)(34,112,55,186)(35,106,56,194)(57,73,64,141)(58,149,65,81)(59,75,66,143)(60,151,67,83)(61,77,68,145)(62,153,69,71)(63,79,70,147)(72,206,154,199)(74,208,142,201)(76,210,144,203)(78,198,146,205)(80,200,148,207)(82,202,150,209)(84,204,152,197)(85,116,92,224)(86,218,93,124)(87,118,94,212)(88,220,95,126)(89,120,96,214)(90,222,97,114)(91,122,98,216)(113,168,221,161)(115,156,223,163)(117,158,211,165)(119,160,213,167)(121,162,215,155)(123,164,217,157)(125,166,219,159), (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), (1,42,12,22)(2,41,13,28)(3,40,14,27)(4,39,8,26)(5,38,9,25)(6,37,10,24)(7,36,11,23)(15,29,43,50)(16,35,44,56)(17,34,45,55)(18,33,46,54)(19,32,47,53)(20,31,48,52)(21,30,49,51)(57,91,64,98)(58,90,65,97)(59,89,66,96)(60,88,67,95)(61,87,68,94)(62,86,69,93)(63,85,70,92)(71,117,153,211)(72,116,154,224)(73,115,141,223)(74,114,142,222)(75,113,143,221)(76,126,144,220)(77,125,145,219)(78,124,146,218)(79,123,147,217)(80,122,148,216)(81,121,149,215)(82,120,150,214)(83,119,151,213)(84,118,152,212)(99,134,187,171)(100,133,188,170)(101,132,189,169)(102,131,190,182)(103,130,191,181)(104,129,192,180)(105,128,193,179)(106,127,194,178)(107,140,195,177)(108,139,196,176)(109,138,183,175)(110,137,184,174)(111,136,185,173)(112,135,186,172)(155,208,162,201)(156,207,163,200)(157,206,164,199)(158,205,165,198)(159,204,166,197)(160,203,167,210)(161,202,168,209) );
G=PermutationGroup([[(1,206,19,63),(2,200,20,57),(3,208,21,65),(4,202,15,59),(5,210,16,67),(6,204,17,61),(7,198,18,69),(8,209,43,66),(9,203,44,60),(10,197,45,68),(11,205,46,62),(12,199,47,70),(13,207,48,64),(14,201,49,58),(22,164,53,85),(23,158,54,93),(24,166,55,87),(25,160,56,95),(26,168,50,89),(27,162,51,97),(28,156,52,91),(29,96,39,161),(30,90,40,155),(31,98,41,163),(32,92,42,157),(33,86,36,165),(34,94,37,159),(35,88,38,167),(71,129,146,136),(72,181,147,174),(73,131,148,138),(74,169,149,176),(75,133,150,140),(76,171,151,178),(77,135,152,128),(78,173,153,180),(79,137,154,130),(80,175,141,182),(81,139,142,132),(82,177,143,170),(83,127,144,134),(84,179,145,172),(99,119,106,220),(100,214,107,113),(101,121,108,222),(102,216,109,115),(103,123,110,224),(104,218,111,117),(105,125,112,212),(114,189,215,196),(116,191,217,184),(118,193,219,186),(120,195,221,188),(122,183,223,190),(124,185,211,192),(126,187,213,194)], [(1,137,12,174),(2,131,13,182),(3,139,14,176),(4,133,8,170),(5,127,9,178),(6,135,10,172),(7,129,11,180),(15,140,43,177),(16,134,44,171),(17,128,45,179),(18,136,46,173),(19,130,47,181),(20,138,48,175),(21,132,49,169),(22,191,42,103),(23,185,36,111),(24,193,37,105),(25,187,38,99),(26,195,39,107),(27,189,40,101),(28,183,41,109),(29,100,50,188),(30,108,51,196),(31,102,52,190),(32,110,53,184),(33,104,54,192),(34,112,55,186),(35,106,56,194),(57,73,64,141),(58,149,65,81),(59,75,66,143),(60,151,67,83),(61,77,68,145),(62,153,69,71),(63,79,70,147),(72,206,154,199),(74,208,142,201),(76,210,144,203),(78,198,146,205),(80,200,148,207),(82,202,150,209),(84,204,152,197),(85,116,92,224),(86,218,93,124),(87,118,94,212),(88,220,95,126),(89,120,96,214),(90,222,97,114),(91,122,98,216),(113,168,221,161),(115,156,223,163),(117,158,211,165),(119,160,213,167),(121,162,215,155),(123,164,217,157),(125,166,219,159)], [(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)], [(1,42,12,22),(2,41,13,28),(3,40,14,27),(4,39,8,26),(5,38,9,25),(6,37,10,24),(7,36,11,23),(15,29,43,50),(16,35,44,56),(17,34,45,55),(18,33,46,54),(19,32,47,53),(20,31,48,52),(21,30,49,51),(57,91,64,98),(58,90,65,97),(59,89,66,96),(60,88,67,95),(61,87,68,94),(62,86,69,93),(63,85,70,92),(71,117,153,211),(72,116,154,224),(73,115,141,223),(74,114,142,222),(75,113,143,221),(76,126,144,220),(77,125,145,219),(78,124,146,218),(79,123,147,217),(80,122,148,216),(81,121,149,215),(82,120,150,214),(83,119,151,213),(84,118,152,212),(99,134,187,171),(100,133,188,170),(101,132,189,169),(102,131,190,182),(103,130,191,181),(104,129,192,180),(105,128,193,179),(106,127,194,178),(107,140,195,177),(108,139,196,176),(109,138,183,175),(110,137,184,174),(111,136,185,173),(112,135,186,172),(155,208,162,201),(156,207,163,200),(157,206,164,199),(158,205,165,198),(159,204,166,197),(160,203,167,210),(161,202,168,209)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4⋊6D14 | D7×C4○D4 | D4.10D14 |
| kernel | C42.137D14 | C4×Dic14 | C42⋊2D7 | C23.11D14 | C22⋊Dic14 | C23.D14 | Dic7⋊4D4 | D14.D4 | Dic7.D4 | C23.18D14 | Dic7⋊D4 | Dic7⋊Q8 | D14⋊3Q8 | C7×C4.4D4 | C4.4D4 | Dic7 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 12 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C42.137D14 ►in GL8(𝔽29)
| 17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 20 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 9 |
| 0 | 0 | 0 | 0 | 9 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 20 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 22 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 19 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 19 |
| 4 | 20 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 25 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 17 |
G:=sub<GL(8,GF(29))| [17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,9,14,0,0,0,0,0,0,15,20,0,0,0,0,20,15,0,0,0,0,0,0,14,9,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[22,19,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,19,19,0,0,0,0,0,0,0,0,22,19,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,19,19],[4,21,0,0,0,0,0,0,20,25,0,0,0,0,0,0,0,0,25,8,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,12,3,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,12,3,0,0,0,0,0,0,0,17] >;
C42.137D14 in GAP, Magma, Sage, TeX
C_4^2._{137}D_{14} % in TeX
G:=Group("C4^2.137D14"); // GroupNames label
G:=SmallGroup(448,1122);
// by ID
G=gap.SmallGroup(448,1122);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,387,100,1123,346,136,18822]);
// 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=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations