metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.178D14, (D4×Dic7)⋊16C2, C4⋊D4.10D7, (C2×D4).152D14, C22⋊C4.47D14, C4.Dic14⋊18C2, Dic7⋊3Q8⋊21C2, C28.201(C4○D4), C4.67(D4⋊2D7), C28.48D4⋊31C2, C28.17D4⋊15C2, (C2×C14).144C24, (C2×C28).501C23, (C22×C4).367D14, C23.11(C22×D7), Dic7.41(C4○D4), (D4×C14).118C22, C22.5(D4⋊2D7), C23.D14⋊14C2, C23.18D14⋊7C2, C23.11D14⋊4C2, Dic7⋊C4.15C22, C4⋊Dic7.205C22, (C22×C14).15C23, (C4×Dic7).91C22, C22.165(C23×D7), C23.D7.21C22, (C22×C28).238C22, C7⋊6(C23.36C23), (C2×Dic7).226C23, (C2×Dic14).152C22, (C22×Dic7).105C22, (C2×C4×Dic7)⋊8C2, C2.35(D7×C4○D4), (C7×C4⋊D4).7C2, C14.149(C2×C4○D4), C2.32(C2×D4⋊2D7), (C2×C14).20(C4○D4), (C7×C4⋊C4).140C22, (C2×C4).292(C22×D7), (C7×C22⋊C4).9C22, SmallGroup(448,1053)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 844 in 234 conjugacy classes, 101 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×8], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C23 [×2], C14 [×3], C14 [×4], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic7 [×2], Dic7 [×7], C28 [×2], C28 [×3], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], Dic14 [×2], C2×Dic7 [×4], C2×Dic7 [×4], C2×Dic7 [×8], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×6], C22×C14, C22×C14 [×2], C23.36C23, C4×Dic7 [×4], C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7, C4⋊Dic7 [×2], C23.D7 [×8], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C22×Dic7 [×2], C22×Dic7 [×2], C22×C28, D4×C14, D4×C14 [×2], C23.11D14 [×2], C23.D14 [×2], Dic7⋊3Q8, C4.Dic14, C2×C4×Dic7, C28.48D4, D4×Dic7, D4×Dic7 [×2], C23.18D14 [×2], C28.17D4, C7×C4⋊D4, C4⋊C4.178D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×6], C24, D14 [×7], C2×C4○D4 [×3], C22×D7 [×7], C23.36C23, D4⋊2D7 [×4], C23×D7, C2×D4⋊2D7 [×2], D7×C4○D4, C4⋊C4.178D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=1, d2=a2b2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c-1 >
►On 224 points
Generators in S224
(1 19 104 120)(2 20 105 121)(3 21 106 122)(4 22 107 123)(5 23 108 124)(6 24 109 125)(7 25 110 126)(8 26 111 113)(9 27 112 114)(10 28 99 115)(11 15 100 116)(12 16 101 117)(13 17 102 118)(14 18 103 119)(29 139 36 132)(30 140 37 133)(31 127 38 134)(32 128 39 135)(33 129 40 136)(34 130 41 137)(35 131 42 138)(43 215 50 222)(44 216 51 223)(45 217 52 224)(46 218 53 211)(47 219 54 212)(48 220 55 213)(49 221 56 214)(57 150 174 156)(58 151 175 157)(59 152 176 158)(60 153 177 159)(61 154 178 160)(62 141 179 161)(63 142 180 162)(64 143 181 163)(65 144 182 164)(66 145 169 165)(67 146 170 166)(68 147 171 167)(69 148 172 168)(70 149 173 155)(71 184 78 191)(72 185 79 192)(73 186 80 193)(74 187 81 194)(75 188 82 195)(76 189 83 196)(77 190 84 183)(85 198 92 205)(86 199 93 206)(87 200 94 207)(88 201 95 208)(89 202 96 209)(90 203 97 210)(91 204 98 197)
(1 133 111 55)(2 56 112 134)(3 135 99 43)(4 44 100 136)(5 137 101 45)(6 46 102 138)(7 139 103 47)(8 48 104 140)(9 127 105 49)(10 50 106 128)(11 129 107 51)(12 52 108 130)(13 131 109 53)(14 54 110 132)(15 33 123 216)(16 217 124 34)(17 35 125 218)(18 219 126 36)(19 37 113 220)(20 221 114 38)(21 39 115 222)(22 223 116 40)(23 41 117 224)(24 211 118 42)(25 29 119 212)(26 213 120 30)(27 31 121 214)(28 215 122 32)(57 76 181 93)(58 94 182 77)(59 78 169 95)(60 96 170 79)(61 80 171 97)(62 98 172 81)(63 82 173 85)(64 86 174 83)(65 84 175 87)(66 88 176 71)(67 72 177 89)(68 90 178 73)(69 74 179 91)(70 92 180 75)(141 204 168 187)(142 188 155 205)(143 206 156 189)(144 190 157 207)(145 208 158 191)(146 192 159 209)(147 210 160 193)(148 194 161 197)(149 198 162 195)(150 196 163 199)(151 200 164 183)(152 184 165 201)(153 202 166 185)(154 186 167 203)
(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)
(1 160 8 167)(2 159 9 166)(3 158 10 165)(4 157 11 164)(5 156 12 163)(6 155 13 162)(7 168 14 161)(15 65 22 58)(16 64 23 57)(17 63 24 70)(18 62 25 69)(19 61 26 68)(20 60 27 67)(21 59 28 66)(29 91 219 81)(30 90 220 80)(31 89 221 79)(32 88 222 78)(33 87 223 77)(34 86 224 76)(35 85 211 75)(36 98 212 74)(37 97 213 73)(38 96 214 72)(39 95 215 71)(40 94 216 84)(41 93 217 83)(42 92 218 82)(43 191 128 201)(44 190 129 200)(45 189 130 199)(46 188 131 198)(47 187 132 197)(48 186 133 210)(49 185 134 209)(50 184 135 208)(51 183 136 207)(52 196 137 206)(53 195 138 205)(54 194 139 204)(55 193 140 203)(56 192 127 202)(99 145 106 152)(100 144 107 151)(101 143 108 150)(102 142 109 149)(103 141 110 148)(104 154 111 147)(105 153 112 146)(113 171 120 178)(114 170 121 177)(115 169 122 176)(116 182 123 175)(117 181 124 174)(118 180 125 173)(119 179 126 172)
G:=sub<Sym(224)| (1,19,104,120)(2,20,105,121)(3,21,106,122)(4,22,107,123)(5,23,108,124)(6,24,109,125)(7,25,110,126)(8,26,111,113)(9,27,112,114)(10,28,99,115)(11,15,100,116)(12,16,101,117)(13,17,102,118)(14,18,103,119)(29,139,36,132)(30,140,37,133)(31,127,38,134)(32,128,39,135)(33,129,40,136)(34,130,41,137)(35,131,42,138)(43,215,50,222)(44,216,51,223)(45,217,52,224)(46,218,53,211)(47,219,54,212)(48,220,55,213)(49,221,56,214)(57,150,174,156)(58,151,175,157)(59,152,176,158)(60,153,177,159)(61,154,178,160)(62,141,179,161)(63,142,180,162)(64,143,181,163)(65,144,182,164)(66,145,169,165)(67,146,170,166)(68,147,171,167)(69,148,172,168)(70,149,173,155)(71,184,78,191)(72,185,79,192)(73,186,80,193)(74,187,81,194)(75,188,82,195)(76,189,83,196)(77,190,84,183)(85,198,92,205)(86,199,93,206)(87,200,94,207)(88,201,95,208)(89,202,96,209)(90,203,97,210)(91,204,98,197), (1,133,111,55)(2,56,112,134)(3,135,99,43)(4,44,100,136)(5,137,101,45)(6,46,102,138)(7,139,103,47)(8,48,104,140)(9,127,105,49)(10,50,106,128)(11,129,107,51)(12,52,108,130)(13,131,109,53)(14,54,110,132)(15,33,123,216)(16,217,124,34)(17,35,125,218)(18,219,126,36)(19,37,113,220)(20,221,114,38)(21,39,115,222)(22,223,116,40)(23,41,117,224)(24,211,118,42)(25,29,119,212)(26,213,120,30)(27,31,121,214)(28,215,122,32)(57,76,181,93)(58,94,182,77)(59,78,169,95)(60,96,170,79)(61,80,171,97)(62,98,172,81)(63,82,173,85)(64,86,174,83)(65,84,175,87)(66,88,176,71)(67,72,177,89)(68,90,178,73)(69,74,179,91)(70,92,180,75)(141,204,168,187)(142,188,155,205)(143,206,156,189)(144,190,157,207)(145,208,158,191)(146,192,159,209)(147,210,160,193)(148,194,161,197)(149,198,162,195)(150,196,163,199)(151,200,164,183)(152,184,165,201)(153,202,166,185)(154,186,167,203), (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), (1,160,8,167)(2,159,9,166)(3,158,10,165)(4,157,11,164)(5,156,12,163)(6,155,13,162)(7,168,14,161)(15,65,22,58)(16,64,23,57)(17,63,24,70)(18,62,25,69)(19,61,26,68)(20,60,27,67)(21,59,28,66)(29,91,219,81)(30,90,220,80)(31,89,221,79)(32,88,222,78)(33,87,223,77)(34,86,224,76)(35,85,211,75)(36,98,212,74)(37,97,213,73)(38,96,214,72)(39,95,215,71)(40,94,216,84)(41,93,217,83)(42,92,218,82)(43,191,128,201)(44,190,129,200)(45,189,130,199)(46,188,131,198)(47,187,132,197)(48,186,133,210)(49,185,134,209)(50,184,135,208)(51,183,136,207)(52,196,137,206)(53,195,138,205)(54,194,139,204)(55,193,140,203)(56,192,127,202)(99,145,106,152)(100,144,107,151)(101,143,108,150)(102,142,109,149)(103,141,110,148)(104,154,111,147)(105,153,112,146)(113,171,120,178)(114,170,121,177)(115,169,122,176)(116,182,123,175)(117,181,124,174)(118,180,125,173)(119,179,126,172)>;
G:=Group( (1,19,104,120)(2,20,105,121)(3,21,106,122)(4,22,107,123)(5,23,108,124)(6,24,109,125)(7,25,110,126)(8,26,111,113)(9,27,112,114)(10,28,99,115)(11,15,100,116)(12,16,101,117)(13,17,102,118)(14,18,103,119)(29,139,36,132)(30,140,37,133)(31,127,38,134)(32,128,39,135)(33,129,40,136)(34,130,41,137)(35,131,42,138)(43,215,50,222)(44,216,51,223)(45,217,52,224)(46,218,53,211)(47,219,54,212)(48,220,55,213)(49,221,56,214)(57,150,174,156)(58,151,175,157)(59,152,176,158)(60,153,177,159)(61,154,178,160)(62,141,179,161)(63,142,180,162)(64,143,181,163)(65,144,182,164)(66,145,169,165)(67,146,170,166)(68,147,171,167)(69,148,172,168)(70,149,173,155)(71,184,78,191)(72,185,79,192)(73,186,80,193)(74,187,81,194)(75,188,82,195)(76,189,83,196)(77,190,84,183)(85,198,92,205)(86,199,93,206)(87,200,94,207)(88,201,95,208)(89,202,96,209)(90,203,97,210)(91,204,98,197), (1,133,111,55)(2,56,112,134)(3,135,99,43)(4,44,100,136)(5,137,101,45)(6,46,102,138)(7,139,103,47)(8,48,104,140)(9,127,105,49)(10,50,106,128)(11,129,107,51)(12,52,108,130)(13,131,109,53)(14,54,110,132)(15,33,123,216)(16,217,124,34)(17,35,125,218)(18,219,126,36)(19,37,113,220)(20,221,114,38)(21,39,115,222)(22,223,116,40)(23,41,117,224)(24,211,118,42)(25,29,119,212)(26,213,120,30)(27,31,121,214)(28,215,122,32)(57,76,181,93)(58,94,182,77)(59,78,169,95)(60,96,170,79)(61,80,171,97)(62,98,172,81)(63,82,173,85)(64,86,174,83)(65,84,175,87)(66,88,176,71)(67,72,177,89)(68,90,178,73)(69,74,179,91)(70,92,180,75)(141,204,168,187)(142,188,155,205)(143,206,156,189)(144,190,157,207)(145,208,158,191)(146,192,159,209)(147,210,160,193)(148,194,161,197)(149,198,162,195)(150,196,163,199)(151,200,164,183)(152,184,165,201)(153,202,166,185)(154,186,167,203), (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), (1,160,8,167)(2,159,9,166)(3,158,10,165)(4,157,11,164)(5,156,12,163)(6,155,13,162)(7,168,14,161)(15,65,22,58)(16,64,23,57)(17,63,24,70)(18,62,25,69)(19,61,26,68)(20,60,27,67)(21,59,28,66)(29,91,219,81)(30,90,220,80)(31,89,221,79)(32,88,222,78)(33,87,223,77)(34,86,224,76)(35,85,211,75)(36,98,212,74)(37,97,213,73)(38,96,214,72)(39,95,215,71)(40,94,216,84)(41,93,217,83)(42,92,218,82)(43,191,128,201)(44,190,129,200)(45,189,130,199)(46,188,131,198)(47,187,132,197)(48,186,133,210)(49,185,134,209)(50,184,135,208)(51,183,136,207)(52,196,137,206)(53,195,138,205)(54,194,139,204)(55,193,140,203)(56,192,127,202)(99,145,106,152)(100,144,107,151)(101,143,108,150)(102,142,109,149)(103,141,110,148)(104,154,111,147)(105,153,112,146)(113,171,120,178)(114,170,121,177)(115,169,122,176)(116,182,123,175)(117,181,124,174)(118,180,125,173)(119,179,126,172) );
G=PermutationGroup([(1,19,104,120),(2,20,105,121),(3,21,106,122),(4,22,107,123),(5,23,108,124),(6,24,109,125),(7,25,110,126),(8,26,111,113),(9,27,112,114),(10,28,99,115),(11,15,100,116),(12,16,101,117),(13,17,102,118),(14,18,103,119),(29,139,36,132),(30,140,37,133),(31,127,38,134),(32,128,39,135),(33,129,40,136),(34,130,41,137),(35,131,42,138),(43,215,50,222),(44,216,51,223),(45,217,52,224),(46,218,53,211),(47,219,54,212),(48,220,55,213),(49,221,56,214),(57,150,174,156),(58,151,175,157),(59,152,176,158),(60,153,177,159),(61,154,178,160),(62,141,179,161),(63,142,180,162),(64,143,181,163),(65,144,182,164),(66,145,169,165),(67,146,170,166),(68,147,171,167),(69,148,172,168),(70,149,173,155),(71,184,78,191),(72,185,79,192),(73,186,80,193),(74,187,81,194),(75,188,82,195),(76,189,83,196),(77,190,84,183),(85,198,92,205),(86,199,93,206),(87,200,94,207),(88,201,95,208),(89,202,96,209),(90,203,97,210),(91,204,98,197)], [(1,133,111,55),(2,56,112,134),(3,135,99,43),(4,44,100,136),(5,137,101,45),(6,46,102,138),(7,139,103,47),(8,48,104,140),(9,127,105,49),(10,50,106,128),(11,129,107,51),(12,52,108,130),(13,131,109,53),(14,54,110,132),(15,33,123,216),(16,217,124,34),(17,35,125,218),(18,219,126,36),(19,37,113,220),(20,221,114,38),(21,39,115,222),(22,223,116,40),(23,41,117,224),(24,211,118,42),(25,29,119,212),(26,213,120,30),(27,31,121,214),(28,215,122,32),(57,76,181,93),(58,94,182,77),(59,78,169,95),(60,96,170,79),(61,80,171,97),(62,98,172,81),(63,82,173,85),(64,86,174,83),(65,84,175,87),(66,88,176,71),(67,72,177,89),(68,90,178,73),(69,74,179,91),(70,92,180,75),(141,204,168,187),(142,188,155,205),(143,206,156,189),(144,190,157,207),(145,208,158,191),(146,192,159,209),(147,210,160,193),(148,194,161,197),(149,198,162,195),(150,196,163,199),(151,200,164,183),(152,184,165,201),(153,202,166,185),(154,186,167,203)], [(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)], [(1,160,8,167),(2,159,9,166),(3,158,10,165),(4,157,11,164),(5,156,12,163),(6,155,13,162),(7,168,14,161),(15,65,22,58),(16,64,23,57),(17,63,24,70),(18,62,25,69),(19,61,26,68),(20,60,27,67),(21,59,28,66),(29,91,219,81),(30,90,220,80),(31,89,221,79),(32,88,222,78),(33,87,223,77),(34,86,224,76),(35,85,211,75),(36,98,212,74),(37,97,213,73),(38,96,214,72),(39,95,215,71),(40,94,216,84),(41,93,217,83),(42,92,218,82),(43,191,128,201),(44,190,129,200),(45,189,130,199),(46,188,131,198),(47,187,132,197),(48,186,133,210),(49,185,134,209),(50,184,135,208),(51,183,136,207),(52,196,137,206),(53,195,138,205),(54,194,139,204),(55,193,140,203),(56,192,127,202),(99,145,106,152),(100,144,107,151),(101,143,108,150),(102,142,109,149),(103,141,110,148),(104,154,111,147),(105,153,112,146),(113,171,120,178),(114,170,121,177),(115,169,122,176),(116,182,123,175),(117,181,124,174),(118,180,125,173),(119,179,126,172)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 8 | 6 | 0 | 0 | 0 | 0 |
| 23 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 23 | 21 | 0 | 0 | 0 | 0 |
| 8 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 10 | 0 | 0 |
| 0 | 0 | 9 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [8,23,0,0,0,0,6,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,17],[23,8,0,0,0,0,21,6,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,17,0,0,0,0,17,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,23,9,0,0,0,0,10,9,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,8,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 4S | 4T | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | D4⋊2D7 | D4⋊2D7 | D7×C4○D4 |
| kernel | C4⋊C4.178D14 | C23.11D14 | C23.D14 | Dic7⋊3Q8 | C4.Dic14 | C2×C4×Dic7 | C28.48D4 | D4×Dic7 | C23.18D14 | C28.17D4 | C7×C4⋊D4 | C4⋊D4 | Dic7 | C28 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 3 | 4 | 4 | 4 | 6 | 3 | 3 | 9 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_4\rtimes C_4._{178}D_{14} % in TeX
G:=Group("C4:C4.178D14"); // GroupNames label
G:=SmallGroup(448,1053);
// by ID
G=gap.SmallGroup(448,1053);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,100,794,297,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=a^2*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
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ℤ
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