?
G = C14.792- (1+4) order 448 = 26·7
34th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.792- (1+4), C28⋊Q8⋊30C2, C4⋊C4.103D14, Dic7.8(C2×D4), C22.42(D4×D7), D14⋊Q8⋊26C2, (C2×D4).158D14, (C2×C28).66C23, C22⋊C4.25D14, (C2×Dic7).79D4, C22.D4⋊2D7, C14.80(C22×D4), (C2×C14).192C24, D14⋊C4.30C22, (C22×C4).254D14, Dic7.D4⋊29C2, C22⋊Dic14⋊27C2, (C22×Dic14)⋊10C2, (D4×C14).130C22, C22.D28⋊18C2, C23.23D14⋊6C2, Dic7⋊C4.37C22, C4⋊Dic7.223C22, (C22×C28).85C22, (C22×C14).28C23, (C22×D7).83C23, C23.198(C22×D7), C22.213(C23×D7), C23.D7.38C22, C23.18D14⋊13C2, C23.11D14⋊10C2, C7⋊3(C23.38C23), (C4×Dic7).119C22, (C2×Dic7).242C23, C2.40(D4.10D14), (C2×Dic14).249C22, (C22×Dic7).126C22, C2.53(C2×D4×D7), (C2×C14).56(C2×D4), (C2×D4⋊2D7).8C2, (C2×C4×D7).108C22, (C7×C4⋊C4).172C22, (C7×C22.D4)⋊2C2, (C2×C4).188(C22×D7), (C2×C7⋊D4).44C22, (C7×C22⋊C4).47C22, SmallGroup(448,1101)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1196 in 270 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×14], C22, C22 [×2], C22 [×8], C7, C2×C4, C2×C4 [×4], C2×C4 [×19], D4 [×6], Q8 [×10], C23 [×2], C23, D7, C14, C14 [×2], C14 [×3], C42 [×2], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8 [×9], C4○D4 [×4], Dic7 [×4], Dic7 [×5], C28 [×5], D14 [×3], C2×C14, C2×C14 [×2], C2×C14 [×5], C42⋊C2, C22⋊Q8 [×4], C22.D4, C22.D4 [×3], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic14 [×10], C4×D7 [×2], C2×Dic7 [×3], C2×Dic7 [×8], C2×Dic7 [×4], C7⋊D4 [×4], C2×C28, C2×C28 [×4], C2×C28 [×2], C7×D4 [×2], C22×D7, C22×C14 [×2], C23.38C23, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7 [×2], D14⋊C4 [×4], C23.D7, C23.D7 [×2], C7×C22⋊C4, C7×C22⋊C4 [×2], C7×C4⋊C4 [×2], C2×Dic14, C2×Dic14 [×4], C2×Dic14 [×4], C2×C4×D7, D4⋊2D7 [×4], C22×Dic7 [×3], C2×C7⋊D4 [×2], C22×C28, D4×C14, C23.11D14, C22⋊Dic14 [×2], Dic7.D4 [×2], C22.D28, C28⋊Q8 [×2], D14⋊Q8 [×2], C23.23D14, C23.18D14, C7×C22.D4, C22×Dic14, C2×D4⋊2D7, C14.792- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2- (1+4) [×2], C22×D7 [×7], C23.38C23, D4×D7 [×2], C23×D7, C2×D4×D7, D4.10D14 [×2], C14.792- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, ebe-1=a7b, cd=dc, ce=ec, ede-1=b2d >
►On 224 points
Generators in S224
(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 86 19 128)(2 87 20 129)(3 88 21 130)(4 89 22 131)(5 90 23 132)(6 91 24 133)(7 92 25 134)(8 93 26 135)(9 94 27 136)(10 95 28 137)(11 96 15 138)(12 97 16 139)(13 98 17 140)(14 85 18 127)(29 173 183 124)(30 174 184 125)(31 175 185 126)(32 176 186 113)(33 177 187 114)(34 178 188 115)(35 179 189 116)(36 180 190 117)(37 181 191 118)(38 182 192 119)(39 169 193 120)(40 170 194 121)(41 171 195 122)(42 172 196 123)(43 103 57 212)(44 104 58 213)(45 105 59 214)(46 106 60 215)(47 107 61 216)(48 108 62 217)(49 109 63 218)(50 110 64 219)(51 111 65 220)(52 112 66 221)(53 99 67 222)(54 100 68 223)(55 101 69 224)(56 102 70 211)(71 143 205 165)(72 144 206 166)(73 145 207 167)(74 146 208 168)(75 147 209 155)(76 148 210 156)(77 149 197 157)(78 150 198 158)(79 151 199 159)(80 152 200 160)(81 153 201 161)(82 154 202 162)(83 141 203 163)(84 142 204 164)
(29 183)(30 184)(31 185)(32 186)(33 187)(34 188)(35 189)(36 190)(37 191)(38 192)(39 193)(40 194)(41 195)(42 196)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 133)(92 134)(93 135)(94 136)(95 137)(96 138)(97 139)(98 140)(141 163)(142 164)(143 165)(144 166)(145 167)(146 168)(147 155)(148 156)(149 157)(150 158)(151 159)(152 160)(153 161)(154 162)
(1 119 19 182)(2 118 20 181)(3 117 21 180)(4 116 22 179)(5 115 23 178)(6 114 24 177)(7 113 25 176)(8 126 26 175)(9 125 27 174)(10 124 28 173)(11 123 15 172)(12 122 16 171)(13 121 17 170)(14 120 18 169)(29 137 183 95)(30 136 184 94)(31 135 185 93)(32 134 186 92)(33 133 187 91)(34 132 188 90)(35 131 189 89)(36 130 190 88)(37 129 191 87)(38 128 192 86)(39 127 193 85)(40 140 194 98)(41 139 195 97)(42 138 196 96)(43 146 57 168)(44 145 58 167)(45 144 59 166)(46 143 60 165)(47 142 61 164)(48 141 62 163)(49 154 63 162)(50 153 64 161)(51 152 65 160)(52 151 66 159)(53 150 67 158)(54 149 68 157)(55 148 69 156)(56 147 70 155)(71 106 205 215)(72 105 206 214)(73 104 207 213)(74 103 208 212)(75 102 209 211)(76 101 210 224)(77 100 197 223)(78 99 198 222)(79 112 199 221)(80 111 200 220)(81 110 201 219)(82 109 202 218)(83 108 203 217)(84 107 204 216)
(1 82 19 202)(2 83 20 203)(3 84 21 204)(4 71 22 205)(5 72 23 206)(6 73 24 207)(7 74 25 208)(8 75 26 209)(9 76 27 210)(10 77 28 197)(11 78 15 198)(12 79 16 199)(13 80 17 200)(14 81 18 201)(29 47 183 61)(30 48 184 62)(31 49 185 63)(32 50 186 64)(33 51 187 65)(34 52 188 66)(35 53 189 67)(36 54 190 68)(37 55 191 69)(38 56 192 70)(39 43 193 57)(40 44 194 58)(41 45 195 59)(42 46 196 60)(85 146 127 168)(86 147 128 155)(87 148 129 156)(88 149 130 157)(89 150 131 158)(90 151 132 159)(91 152 133 160)(92 153 134 161)(93 154 135 162)(94 141 136 163)(95 142 137 164)(96 143 138 165)(97 144 139 166)(98 145 140 167)(99 123 222 172)(100 124 223 173)(101 125 224 174)(102 126 211 175)(103 113 212 176)(104 114 213 177)(105 115 214 178)(106 116 215 179)(107 117 216 180)(108 118 217 181)(109 119 218 182)(110 120 219 169)(111 121 220 170)(112 122 221 171)
G:=sub<Sym(224)| (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), (1,86,19,128)(2,87,20,129)(3,88,21,130)(4,89,22,131)(5,90,23,132)(6,91,24,133)(7,92,25,134)(8,93,26,135)(9,94,27,136)(10,95,28,137)(11,96,15,138)(12,97,16,139)(13,98,17,140)(14,85,18,127)(29,173,183,124)(30,174,184,125)(31,175,185,126)(32,176,186,113)(33,177,187,114)(34,178,188,115)(35,179,189,116)(36,180,190,117)(37,181,191,118)(38,182,192,119)(39,169,193,120)(40,170,194,121)(41,171,195,122)(42,172,196,123)(43,103,57,212)(44,104,58,213)(45,105,59,214)(46,106,60,215)(47,107,61,216)(48,108,62,217)(49,109,63,218)(50,110,64,219)(51,111,65,220)(52,112,66,221)(53,99,67,222)(54,100,68,223)(55,101,69,224)(56,102,70,211)(71,143,205,165)(72,144,206,166)(73,145,207,167)(74,146,208,168)(75,147,209,155)(76,148,210,156)(77,149,197,157)(78,150,198,158)(79,151,199,159)(80,152,200,160)(81,153,201,161)(82,154,202,162)(83,141,203,163)(84,142,204,164), (29,183)(30,184)(31,185)(32,186)(33,187)(34,188)(35,189)(36,190)(37,191)(38,192)(39,193)(40,194)(41,195)(42,196)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,133)(92,134)(93,135)(94,136)(95,137)(96,138)(97,139)(98,140)(141,163)(142,164)(143,165)(144,166)(145,167)(146,168)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(153,161)(154,162), (1,119,19,182)(2,118,20,181)(3,117,21,180)(4,116,22,179)(5,115,23,178)(6,114,24,177)(7,113,25,176)(8,126,26,175)(9,125,27,174)(10,124,28,173)(11,123,15,172)(12,122,16,171)(13,121,17,170)(14,120,18,169)(29,137,183,95)(30,136,184,94)(31,135,185,93)(32,134,186,92)(33,133,187,91)(34,132,188,90)(35,131,189,89)(36,130,190,88)(37,129,191,87)(38,128,192,86)(39,127,193,85)(40,140,194,98)(41,139,195,97)(42,138,196,96)(43,146,57,168)(44,145,58,167)(45,144,59,166)(46,143,60,165)(47,142,61,164)(48,141,62,163)(49,154,63,162)(50,153,64,161)(51,152,65,160)(52,151,66,159)(53,150,67,158)(54,149,68,157)(55,148,69,156)(56,147,70,155)(71,106,205,215)(72,105,206,214)(73,104,207,213)(74,103,208,212)(75,102,209,211)(76,101,210,224)(77,100,197,223)(78,99,198,222)(79,112,199,221)(80,111,200,220)(81,110,201,219)(82,109,202,218)(83,108,203,217)(84,107,204,216), (1,82,19,202)(2,83,20,203)(3,84,21,204)(4,71,22,205)(5,72,23,206)(6,73,24,207)(7,74,25,208)(8,75,26,209)(9,76,27,210)(10,77,28,197)(11,78,15,198)(12,79,16,199)(13,80,17,200)(14,81,18,201)(29,47,183,61)(30,48,184,62)(31,49,185,63)(32,50,186,64)(33,51,187,65)(34,52,188,66)(35,53,189,67)(36,54,190,68)(37,55,191,69)(38,56,192,70)(39,43,193,57)(40,44,194,58)(41,45,195,59)(42,46,196,60)(85,146,127,168)(86,147,128,155)(87,148,129,156)(88,149,130,157)(89,150,131,158)(90,151,132,159)(91,152,133,160)(92,153,134,161)(93,154,135,162)(94,141,136,163)(95,142,137,164)(96,143,138,165)(97,144,139,166)(98,145,140,167)(99,123,222,172)(100,124,223,173)(101,125,224,174)(102,126,211,175)(103,113,212,176)(104,114,213,177)(105,115,214,178)(106,116,215,179)(107,117,216,180)(108,118,217,181)(109,119,218,182)(110,120,219,169)(111,121,220,170)(112,122,221,171)>;
G:=Group( (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), (1,86,19,128)(2,87,20,129)(3,88,21,130)(4,89,22,131)(5,90,23,132)(6,91,24,133)(7,92,25,134)(8,93,26,135)(9,94,27,136)(10,95,28,137)(11,96,15,138)(12,97,16,139)(13,98,17,140)(14,85,18,127)(29,173,183,124)(30,174,184,125)(31,175,185,126)(32,176,186,113)(33,177,187,114)(34,178,188,115)(35,179,189,116)(36,180,190,117)(37,181,191,118)(38,182,192,119)(39,169,193,120)(40,170,194,121)(41,171,195,122)(42,172,196,123)(43,103,57,212)(44,104,58,213)(45,105,59,214)(46,106,60,215)(47,107,61,216)(48,108,62,217)(49,109,63,218)(50,110,64,219)(51,111,65,220)(52,112,66,221)(53,99,67,222)(54,100,68,223)(55,101,69,224)(56,102,70,211)(71,143,205,165)(72,144,206,166)(73,145,207,167)(74,146,208,168)(75,147,209,155)(76,148,210,156)(77,149,197,157)(78,150,198,158)(79,151,199,159)(80,152,200,160)(81,153,201,161)(82,154,202,162)(83,141,203,163)(84,142,204,164), (29,183)(30,184)(31,185)(32,186)(33,187)(34,188)(35,189)(36,190)(37,191)(38,192)(39,193)(40,194)(41,195)(42,196)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,133)(92,134)(93,135)(94,136)(95,137)(96,138)(97,139)(98,140)(141,163)(142,164)(143,165)(144,166)(145,167)(146,168)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(153,161)(154,162), (1,119,19,182)(2,118,20,181)(3,117,21,180)(4,116,22,179)(5,115,23,178)(6,114,24,177)(7,113,25,176)(8,126,26,175)(9,125,27,174)(10,124,28,173)(11,123,15,172)(12,122,16,171)(13,121,17,170)(14,120,18,169)(29,137,183,95)(30,136,184,94)(31,135,185,93)(32,134,186,92)(33,133,187,91)(34,132,188,90)(35,131,189,89)(36,130,190,88)(37,129,191,87)(38,128,192,86)(39,127,193,85)(40,140,194,98)(41,139,195,97)(42,138,196,96)(43,146,57,168)(44,145,58,167)(45,144,59,166)(46,143,60,165)(47,142,61,164)(48,141,62,163)(49,154,63,162)(50,153,64,161)(51,152,65,160)(52,151,66,159)(53,150,67,158)(54,149,68,157)(55,148,69,156)(56,147,70,155)(71,106,205,215)(72,105,206,214)(73,104,207,213)(74,103,208,212)(75,102,209,211)(76,101,210,224)(77,100,197,223)(78,99,198,222)(79,112,199,221)(80,111,200,220)(81,110,201,219)(82,109,202,218)(83,108,203,217)(84,107,204,216), (1,82,19,202)(2,83,20,203)(3,84,21,204)(4,71,22,205)(5,72,23,206)(6,73,24,207)(7,74,25,208)(8,75,26,209)(9,76,27,210)(10,77,28,197)(11,78,15,198)(12,79,16,199)(13,80,17,200)(14,81,18,201)(29,47,183,61)(30,48,184,62)(31,49,185,63)(32,50,186,64)(33,51,187,65)(34,52,188,66)(35,53,189,67)(36,54,190,68)(37,55,191,69)(38,56,192,70)(39,43,193,57)(40,44,194,58)(41,45,195,59)(42,46,196,60)(85,146,127,168)(86,147,128,155)(87,148,129,156)(88,149,130,157)(89,150,131,158)(90,151,132,159)(91,152,133,160)(92,153,134,161)(93,154,135,162)(94,141,136,163)(95,142,137,164)(96,143,138,165)(97,144,139,166)(98,145,140,167)(99,123,222,172)(100,124,223,173)(101,125,224,174)(102,126,211,175)(103,113,212,176)(104,114,213,177)(105,115,214,178)(106,116,215,179)(107,117,216,180)(108,118,217,181)(109,119,218,182)(110,120,219,169)(111,121,220,170)(112,122,221,171) );
G=PermutationGroup([(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)], [(1,86,19,128),(2,87,20,129),(3,88,21,130),(4,89,22,131),(5,90,23,132),(6,91,24,133),(7,92,25,134),(8,93,26,135),(9,94,27,136),(10,95,28,137),(11,96,15,138),(12,97,16,139),(13,98,17,140),(14,85,18,127),(29,173,183,124),(30,174,184,125),(31,175,185,126),(32,176,186,113),(33,177,187,114),(34,178,188,115),(35,179,189,116),(36,180,190,117),(37,181,191,118),(38,182,192,119),(39,169,193,120),(40,170,194,121),(41,171,195,122),(42,172,196,123),(43,103,57,212),(44,104,58,213),(45,105,59,214),(46,106,60,215),(47,107,61,216),(48,108,62,217),(49,109,63,218),(50,110,64,219),(51,111,65,220),(52,112,66,221),(53,99,67,222),(54,100,68,223),(55,101,69,224),(56,102,70,211),(71,143,205,165),(72,144,206,166),(73,145,207,167),(74,146,208,168),(75,147,209,155),(76,148,210,156),(77,149,197,157),(78,150,198,158),(79,151,199,159),(80,152,200,160),(81,153,201,161),(82,154,202,162),(83,141,203,163),(84,142,204,164)], [(29,183),(30,184),(31,185),(32,186),(33,187),(34,188),(35,189),(36,190),(37,191),(38,192),(39,193),(40,194),(41,195),(42,196),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70),(85,127),(86,128),(87,129),(88,130),(89,131),(90,132),(91,133),(92,134),(93,135),(94,136),(95,137),(96,138),(97,139),(98,140),(141,163),(142,164),(143,165),(144,166),(145,167),(146,168),(147,155),(148,156),(149,157),(150,158),(151,159),(152,160),(153,161),(154,162)], [(1,119,19,182),(2,118,20,181),(3,117,21,180),(4,116,22,179),(5,115,23,178),(6,114,24,177),(7,113,25,176),(8,126,26,175),(9,125,27,174),(10,124,28,173),(11,123,15,172),(12,122,16,171),(13,121,17,170),(14,120,18,169),(29,137,183,95),(30,136,184,94),(31,135,185,93),(32,134,186,92),(33,133,187,91),(34,132,188,90),(35,131,189,89),(36,130,190,88),(37,129,191,87),(38,128,192,86),(39,127,193,85),(40,140,194,98),(41,139,195,97),(42,138,196,96),(43,146,57,168),(44,145,58,167),(45,144,59,166),(46,143,60,165),(47,142,61,164),(48,141,62,163),(49,154,63,162),(50,153,64,161),(51,152,65,160),(52,151,66,159),(53,150,67,158),(54,149,68,157),(55,148,69,156),(56,147,70,155),(71,106,205,215),(72,105,206,214),(73,104,207,213),(74,103,208,212),(75,102,209,211),(76,101,210,224),(77,100,197,223),(78,99,198,222),(79,112,199,221),(80,111,200,220),(81,110,201,219),(82,109,202,218),(83,108,203,217),(84,107,204,216)], [(1,82,19,202),(2,83,20,203),(3,84,21,204),(4,71,22,205),(5,72,23,206),(6,73,24,207),(7,74,25,208),(8,75,26,209),(9,76,27,210),(10,77,28,197),(11,78,15,198),(12,79,16,199),(13,80,17,200),(14,81,18,201),(29,47,183,61),(30,48,184,62),(31,49,185,63),(32,50,186,64),(33,51,187,65),(34,52,188,66),(35,53,189,67),(36,54,190,68),(37,55,191,69),(38,56,192,70),(39,43,193,57),(40,44,194,58),(41,45,195,59),(42,46,196,60),(85,146,127,168),(86,147,128,155),(87,148,129,156),(88,149,130,157),(89,150,131,158),(90,151,132,159),(91,152,133,160),(92,153,134,161),(93,154,135,162),(94,141,136,163),(95,142,137,164),(96,143,138,165),(97,144,139,166),(98,145,140,167),(99,123,222,172),(100,124,223,173),(101,125,224,174),(102,126,211,175),(103,113,212,176),(104,114,213,177),(105,115,214,178),(106,116,215,179),(107,117,216,180),(108,118,217,181),(109,119,218,182),(110,120,219,169),(111,121,220,170),(112,122,221,171)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 10 | 0 | 0 |
| 0 | 0 | 14 | 28 | 0 | 0 |
| 0 | 0 | 3 | 28 | 4 | 10 |
| 0 | 0 | 26 | 26 | 14 | 28 |
| 26 | 22 | 0 | 0 | 0 | 0 |
| 26 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 25 | 2 | 0 |
| 0 | 0 | 24 | 10 | 0 | 2 |
| 0 | 0 | 22 | 19 | 15 | 4 |
| 0 | 0 | 2 | 12 | 5 | 19 |
| 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 | 15 | 4 | 28 | 0 |
| 0 | 0 | 5 | 19 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 8 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 11 | 12 | 22 | 8 |
| 0 | 0 | 25 | 18 | 1 | 7 |
| 28 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 23 | 0 | 0 |
| 0 | 0 | 9 | 16 | 0 | 0 |
| 0 | 0 | 4 | 18 | 13 | 23 |
| 0 | 0 | 25 | 25 | 9 | 16 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,4,14,3,26,0,0,10,28,28,26,0,0,0,0,4,14,0,0,0,0,10,28],[26,26,0,0,0,0,22,3,0,0,0,0,0,0,14,24,22,2,0,0,25,10,19,12,0,0,2,0,15,5,0,0,0,2,4,19],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,15,5,0,0,0,1,4,19,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,22,1,11,25,0,0,8,7,12,18,0,0,0,0,22,1,0,0,0,0,8,7],[28,0,0,0,0,0,2,1,0,0,0,0,0,0,13,9,4,25,0,0,23,16,18,25,0,0,0,0,13,9,0,0,0,0,23,16] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 28 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2- (1+4) | D4×D7 | D4.10D14 |
| kernel | C14.792- (1+4) | C23.11D14 | C22⋊Dic14 | Dic7.D4 | C22.D28 | C28⋊Q8 | D14⋊Q8 | C23.23D14 | C23.18D14 | C7×C22.D4 | C22×Dic14 | C2×D4⋊2D7 | C2×Dic7 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 9 | 6 | 3 | 3 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_{14}._{79}2_-^{(1+4)} % in TeX
G:=Group("C14.79ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1101);
// by ID
G=gap.SmallGroup(448,1101);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,387,1123,185,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀