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G = C14.652+ (1+4)  order 448 = 26·7

65th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C14.652+ (1+4), C14.842- (1+4), C28⋊Q8⋊33C2, C4⋊C4.199D14, D14⋊Q8⋊32C2, (C2×D4).102D14, C22⋊C4.29D14, Dic7⋊3Q8⋊32C2, D14.D4⋊35C2, C28.48D4⋊23C2, C28.17D4⋊22C2, (C2×C14).206C24, (C2×C28).181C23, D14⋊C4.34C22, Dic7.9(C4○D4), Dic7⋊D4.2C2, (C22×C4).260D14, C22.D4⋊11D7, C4⋊Dic7.49C22, C2.67(D4⋊6D14), C23.30(C22×D7), Dic7.D4⋊34C2, C22⋊Dic14⋊34C2, (D4×C14).144C22, C23.D14⋊34C2, Dic7⋊C4.44C22, (C22×C14).38C23, (C22×D7).87C23, C22.227(C23×D7), C23.D7.45C22, C23.11D14⋊15C2, (C22×C28).116C22, C7⋊7(C22.36C24), (C2×Dic7).107C23, (C4×Dic7).125C22, C2.45(D4.10D14), (C2×Dic14).169C22, (C22×Dic7).132C22, (C4×C7⋊D4)⋊8C2, C2.68(D7×C4○D4), C4⋊C4⋊D7⋊30C2, C14.180(C2×C4○D4), (C2×C4×D7).212C22, (C2×C4).68(C22×D7), (C7×C4⋊C4).179C22, (C2×C7⋊D4).50C22, (C7×C22.D4)⋊14C2, (C7×C22⋊C4).54C22, SmallGroup(448,1115)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C14.652+ (1+4)
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D14.D4 — C14.652+ (1+4)
Lower central
C7 — C2×C14 — C14.652+ (1+4)

Subgroups: 940 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×13], C22, C22 [×9], C7, C2×C4 [×5], C2×C4 [×11], D4 [×4], Q8 [×4], C23 [×2], C23, D7, C14 [×3], C14 [×2], C42 [×4], C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8 [×3], Dic7 [×2], Dic7 [×6], C28 [×5], D14 [×3], C2×C14, C2×C14 [×6], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4, C22.D4, C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, Dic14 [×4], C4×D7, C2×Dic7 [×7], C2×Dic7 [×2], C7⋊D4 [×3], C2×C28 [×5], C2×C28, C7×D4, C22×D7, C22×C14 [×2], C22.36C24, C4×Dic7 [×4], Dic7⋊C4 [×6], C4⋊Dic7 [×2], D14⋊C4 [×4], C23.D7 [×5], C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×Dic14 [×3], C2×C4×D7, C22×Dic7, C2×C7⋊D4 [×2], C22×C28, D4×C14, C23.11D14, C22⋊Dic14, C23.D14, D14.D4, Dic7.D4 [×2], Dic7⋊3Q8, C28⋊Q8, D14⋊Q8, C4⋊C4⋊D7, C28.48D4, C4×C7⋊D4, C28.17D4, Dic7⋊D4, C7×C22.D4, C14.652+ (1+4)

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D7 [×7], C22.36C24, C23×D7, D4⋊6D14, D7×C4○D4, D4.10D14, C14.652+ (1+4)

Generators and relations
 G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=a7b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a7b-1, bd=db, ebe=a7b, cd=dc, ce=ec, ede=a7b2d >

Smallest permutation representation
►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)

(1 106 17 88)(2 107 18 89)(3 108 19 90)(4 109 20 91)(5 110 21 92)(6 111 22 93)(7 112 23 94)(8 99 24 95)(9 100 25 96)(10 101 26 97)(11 102 27 98)(12 103 28 85)(13 104 15 86)(14 105 16 87)(29 78 53 62)(30 79 54 63)(31 80 55 64)(32 81 56 65)(33 82 43 66)(34 83 44 67)(35 84 45 68)(36 71 46 69)(37 72 47 70)(38 73 48 57)(39 74 49 58)(40 75 50 59)(41 76 51 60)(42 77 52 61)(113 211 134 202)(114 212 135 203)(115 213 136 204)(116 214 137 205)(117 215 138 206)(118 216 139 207)(119 217 140 208)(120 218 127 209)(121 219 128 210)(122 220 129 197)(123 221 130 198)(124 222 131 199)(125 223 132 200)(126 224 133 201)(141 183 165 175)(142 184 166 176)(143 185 167 177)(144 186 168 178)(145 187 155 179)(146 188 156 180)(147 189 157 181)(148 190 158 182)(149 191 159 169)(150 192 160 170)(151 193 161 171)(152 194 162 172)(153 195 163 173)(154 196 164 174)

(1 70 8 63)(2 57 9 64)(3 58 10 65)(4 59 11 66)(5 60 12 67)(6 61 13 68)(7 62 14 69)(15 84 22 77)(16 71 23 78)(17 72 24 79)(18 73 25 80)(19 74 26 81)(20 75 27 82)(21 76 28 83)(29 94 36 87)(30 95 37 88)(31 96 38 89)(32 97 39 90)(33 98 40 91)(34 85 41 92)(35 86 42 93)(43 102 50 109)(44 103 51 110)(45 104 52 111)(46 105 53 112)(47 106 54 99)(48 107 55 100)(49 108 56 101)(113 171 120 178)(114 172 121 179)(115 173 122 180)(116 174 123 181)(117 175 124 182)(118 176 125 169)(119 177 126 170)(127 186 134 193)(128 187 135 194)(129 188 136 195)(130 189 137 196)(131 190 138 183)(132 191 139 184)(133 192 140 185)(141 206 148 199)(142 207 149 200)(143 208 150 201)(144 209 151 202)(145 210 152 203)(146 197 153 204)(147 198 154 205)(155 219 162 212)(156 220 163 213)(157 221 164 214)(158 222 165 215)(159 223 166 216)(160 224 167 217)(161 211 168 218)

(1 126 24 140)(2 125 25 139)(3 124 26 138)(4 123 27 137)(5 122 28 136)(6 121 15 135)(7 120 16 134)(8 119 17 133)(9 118 18 132)(10 117 19 131)(11 116 20 130)(12 115 21 129)(13 114 22 128)(14 113 23 127)(29 144 46 161)(30 143 47 160)(31 142 48 159)(32 141 49 158)(33 154 50 157)(34 153 51 156)(35 152 52 155)(36 151 53 168)(37 150 54 167)(38 149 55 166)(39 148 56 165)(40 147 43 164)(41 146 44 163)(42 145 45 162)(57 169 80 184)(58 182 81 183)(59 181 82 196)(60 180 83 195)(61 179 84 194)(62 178 71 193)(63 177 72 192)(64 176 73 191)(65 175 74 190)(66 174 75 189)(67 173 76 188)(68 172 77 187)(69 171 78 186)(70 170 79 185)(85 204 110 220)(86 203 111 219)(87 202 112 218)(88 201 99 217)(89 200 100 216)(90 199 101 215)(91 198 102 214)(92 197 103 213)(93 210 104 212)(94 209 105 211)(95 208 106 224)(96 207 107 223)(97 206 108 222)(98 205 109 221)

(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 165)(142 166)(143 167)(144 168)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)(151 161)(152 162)(153 163)(154 164)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 190)(176 191)(177 192)(178 193)(179 194)(180 195)(181 196)(182 183)(197 220)(198 221)(199 222)(200 223)(201 224)(202 211)(203 212)(204 213)(205 214)(206 215)(207 216)(208 217)(209 218)(210 219)
 
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), (1,106,17,88)(2,107,18,89)(3,108,19,90)(4,109,20,91)(5,110,21,92)(6,111,22,93)(7,112,23,94)(8,99,24,95)(9,100,25,96)(10,101,26,97)(11,102,27,98)(12,103,28,85)(13,104,15,86)(14,105,16,87)(29,78,53,62)(30,79,54,63)(31,80,55,64)(32,81,56,65)(33,82,43,66)(34,83,44,67)(35,84,45,68)(36,71,46,69)(37,72,47,70)(38,73,48,57)(39,74,49,58)(40,75,50,59)(41,76,51,60)(42,77,52,61)(113,211,134,202)(114,212,135,203)(115,213,136,204)(116,214,137,205)(117,215,138,206)(118,216,139,207)(119,217,140,208)(120,218,127,209)(121,219,128,210)(122,220,129,197)(123,221,130,198)(124,222,131,199)(125,223,132,200)(126,224,133,201)(141,183,165,175)(142,184,166,176)(143,185,167,177)(144,186,168,178)(145,187,155,179)(146,188,156,180)(147,189,157,181)(148,190,158,182)(149,191,159,169)(150,192,160,170)(151,193,161,171)(152,194,162,172)(153,195,163,173)(154,196,164,174), (1,70,8,63)(2,57,9,64)(3,58,10,65)(4,59,11,66)(5,60,12,67)(6,61,13,68)(7,62,14,69)(15,84,22,77)(16,71,23,78)(17,72,24,79)(18,73,25,80)(19,74,26,81)(20,75,27,82)(21,76,28,83)(29,94,36,87)(30,95,37,88)(31,96,38,89)(32,97,39,90)(33,98,40,91)(34,85,41,92)(35,86,42,93)(43,102,50,109)(44,103,51,110)(45,104,52,111)(46,105,53,112)(47,106,54,99)(48,107,55,100)(49,108,56,101)(113,171,120,178)(114,172,121,179)(115,173,122,180)(116,174,123,181)(117,175,124,182)(118,176,125,169)(119,177,126,170)(127,186,134,193)(128,187,135,194)(129,188,136,195)(130,189,137,196)(131,190,138,183)(132,191,139,184)(133,192,140,185)(141,206,148,199)(142,207,149,200)(143,208,150,201)(144,209,151,202)(145,210,152,203)(146,197,153,204)(147,198,154,205)(155,219,162,212)(156,220,163,213)(157,221,164,214)(158,222,165,215)(159,223,166,216)(160,224,167,217)(161,211,168,218), (1,126,24,140)(2,125,25,139)(3,124,26,138)(4,123,27,137)(5,122,28,136)(6,121,15,135)(7,120,16,134)(8,119,17,133)(9,118,18,132)(10,117,19,131)(11,116,20,130)(12,115,21,129)(13,114,22,128)(14,113,23,127)(29,144,46,161)(30,143,47,160)(31,142,48,159)(32,141,49,158)(33,154,50,157)(34,153,51,156)(35,152,52,155)(36,151,53,168)(37,150,54,167)(38,149,55,166)(39,148,56,165)(40,147,43,164)(41,146,44,163)(42,145,45,162)(57,169,80,184)(58,182,81,183)(59,181,82,196)(60,180,83,195)(61,179,84,194)(62,178,71,193)(63,177,72,192)(64,176,73,191)(65,175,74,190)(66,174,75,189)(67,173,76,188)(68,172,77,187)(69,171,78,186)(70,170,79,185)(85,204,110,220)(86,203,111,219)(87,202,112,218)(88,201,99,217)(89,200,100,216)(90,199,101,215)(91,198,102,214)(92,197,103,213)(93,210,104,212)(94,209,105,211)(95,208,106,224)(96,207,107,223)(97,206,108,222)(98,205,109,221), (29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,165)(142,166)(143,167)(144,168)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195)(181,196)(182,183)(197,220)(198,221)(199,222)(200,223)(201,224)(202,211)(203,212)(204,213)(205,214)(206,215)(207,216)(208,217)(209,218)(210,219)>;
 
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), (1,106,17,88)(2,107,18,89)(3,108,19,90)(4,109,20,91)(5,110,21,92)(6,111,22,93)(7,112,23,94)(8,99,24,95)(9,100,25,96)(10,101,26,97)(11,102,27,98)(12,103,28,85)(13,104,15,86)(14,105,16,87)(29,78,53,62)(30,79,54,63)(31,80,55,64)(32,81,56,65)(33,82,43,66)(34,83,44,67)(35,84,45,68)(36,71,46,69)(37,72,47,70)(38,73,48,57)(39,74,49,58)(40,75,50,59)(41,76,51,60)(42,77,52,61)(113,211,134,202)(114,212,135,203)(115,213,136,204)(116,214,137,205)(117,215,138,206)(118,216,139,207)(119,217,140,208)(120,218,127,209)(121,219,128,210)(122,220,129,197)(123,221,130,198)(124,222,131,199)(125,223,132,200)(126,224,133,201)(141,183,165,175)(142,184,166,176)(143,185,167,177)(144,186,168,178)(145,187,155,179)(146,188,156,180)(147,189,157,181)(148,190,158,182)(149,191,159,169)(150,192,160,170)(151,193,161,171)(152,194,162,172)(153,195,163,173)(154,196,164,174), (1,70,8,63)(2,57,9,64)(3,58,10,65)(4,59,11,66)(5,60,12,67)(6,61,13,68)(7,62,14,69)(15,84,22,77)(16,71,23,78)(17,72,24,79)(18,73,25,80)(19,74,26,81)(20,75,27,82)(21,76,28,83)(29,94,36,87)(30,95,37,88)(31,96,38,89)(32,97,39,90)(33,98,40,91)(34,85,41,92)(35,86,42,93)(43,102,50,109)(44,103,51,110)(45,104,52,111)(46,105,53,112)(47,106,54,99)(48,107,55,100)(49,108,56,101)(113,171,120,178)(114,172,121,179)(115,173,122,180)(116,174,123,181)(117,175,124,182)(118,176,125,169)(119,177,126,170)(127,186,134,193)(128,187,135,194)(129,188,136,195)(130,189,137,196)(131,190,138,183)(132,191,139,184)(133,192,140,185)(141,206,148,199)(142,207,149,200)(143,208,150,201)(144,209,151,202)(145,210,152,203)(146,197,153,204)(147,198,154,205)(155,219,162,212)(156,220,163,213)(157,221,164,214)(158,222,165,215)(159,223,166,216)(160,224,167,217)(161,211,168,218), (1,126,24,140)(2,125,25,139)(3,124,26,138)(4,123,27,137)(5,122,28,136)(6,121,15,135)(7,120,16,134)(8,119,17,133)(9,118,18,132)(10,117,19,131)(11,116,20,130)(12,115,21,129)(13,114,22,128)(14,113,23,127)(29,144,46,161)(30,143,47,160)(31,142,48,159)(32,141,49,158)(33,154,50,157)(34,153,51,156)(35,152,52,155)(36,151,53,168)(37,150,54,167)(38,149,55,166)(39,148,56,165)(40,147,43,164)(41,146,44,163)(42,145,45,162)(57,169,80,184)(58,182,81,183)(59,181,82,196)(60,180,83,195)(61,179,84,194)(62,178,71,193)(63,177,72,192)(64,176,73,191)(65,175,74,190)(66,174,75,189)(67,173,76,188)(68,172,77,187)(69,171,78,186)(70,170,79,185)(85,204,110,220)(86,203,111,219)(87,202,112,218)(88,201,99,217)(89,200,100,216)(90,199,101,215)(91,198,102,214)(92,197,103,213)(93,210,104,212)(94,209,105,211)(95,208,106,224)(96,207,107,223)(97,206,108,222)(98,205,109,221), (29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,165)(142,166)(143,167)(144,168)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195)(181,196)(182,183)(197,220)(198,221)(199,222)(200,223)(201,224)(202,211)(203,212)(204,213)(205,214)(206,215)(207,216)(208,217)(209,218)(210,219) );
 
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)], [(1,106,17,88),(2,107,18,89),(3,108,19,90),(4,109,20,91),(5,110,21,92),(6,111,22,93),(7,112,23,94),(8,99,24,95),(9,100,25,96),(10,101,26,97),(11,102,27,98),(12,103,28,85),(13,104,15,86),(14,105,16,87),(29,78,53,62),(30,79,54,63),(31,80,55,64),(32,81,56,65),(33,82,43,66),(34,83,44,67),(35,84,45,68),(36,71,46,69),(37,72,47,70),(38,73,48,57),(39,74,49,58),(40,75,50,59),(41,76,51,60),(42,77,52,61),(113,211,134,202),(114,212,135,203),(115,213,136,204),(116,214,137,205),(117,215,138,206),(118,216,139,207),(119,217,140,208),(120,218,127,209),(121,219,128,210),(122,220,129,197),(123,221,130,198),(124,222,131,199),(125,223,132,200),(126,224,133,201),(141,183,165,175),(142,184,166,176),(143,185,167,177),(144,186,168,178),(145,187,155,179),(146,188,156,180),(147,189,157,181),(148,190,158,182),(149,191,159,169),(150,192,160,170),(151,193,161,171),(152,194,162,172),(153,195,163,173),(154,196,164,174)], [(1,70,8,63),(2,57,9,64),(3,58,10,65),(4,59,11,66),(5,60,12,67),(6,61,13,68),(7,62,14,69),(15,84,22,77),(16,71,23,78),(17,72,24,79),(18,73,25,80),(19,74,26,81),(20,75,27,82),(21,76,28,83),(29,94,36,87),(30,95,37,88),(31,96,38,89),(32,97,39,90),(33,98,40,91),(34,85,41,92),(35,86,42,93),(43,102,50,109),(44,103,51,110),(45,104,52,111),(46,105,53,112),(47,106,54,99),(48,107,55,100),(49,108,56,101),(113,171,120,178),(114,172,121,179),(115,173,122,180),(116,174,123,181),(117,175,124,182),(118,176,125,169),(119,177,126,170),(127,186,134,193),(128,187,135,194),(129,188,136,195),(130,189,137,196),(131,190,138,183),(132,191,139,184),(133,192,140,185),(141,206,148,199),(142,207,149,200),(143,208,150,201),(144,209,151,202),(145,210,152,203),(146,197,153,204),(147,198,154,205),(155,219,162,212),(156,220,163,213),(157,221,164,214),(158,222,165,215),(159,223,166,216),(160,224,167,217),(161,211,168,218)], [(1,126,24,140),(2,125,25,139),(3,124,26,138),(4,123,27,137),(5,122,28,136),(6,121,15,135),(7,120,16,134),(8,119,17,133),(9,118,18,132),(10,117,19,131),(11,116,20,130),(12,115,21,129),(13,114,22,128),(14,113,23,127),(29,144,46,161),(30,143,47,160),(31,142,48,159),(32,141,49,158),(33,154,50,157),(34,153,51,156),(35,152,52,155),(36,151,53,168),(37,150,54,167),(38,149,55,166),(39,148,56,165),(40,147,43,164),(41,146,44,163),(42,145,45,162),(57,169,80,184),(58,182,81,183),(59,181,82,196),(60,180,83,195),(61,179,84,194),(62,178,71,193),(63,177,72,192),(64,176,73,191),(65,175,74,190),(66,174,75,189),(67,173,76,188),(68,172,77,187),(69,171,78,186),(70,170,79,185),(85,204,110,220),(86,203,111,219),(87,202,112,218),(88,201,99,217),(89,200,100,216),(90,199,101,215),(91,198,102,214),(92,197,103,213),(93,210,104,212),(94,209,105,211),(95,208,106,224),(96,207,107,223),(97,206,108,222),(98,205,109,221)], [(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,165),(142,166),(143,167),(144,168),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160),(151,161),(152,162),(153,163),(154,164),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,190),(176,191),(177,192),(178,193),(179,194),(180,195),(181,196),(182,183),(197,220),(198,221),(199,222),(200,223),(201,224),(202,211),(203,212),(204,213),(205,214),(206,215),(207,216),(208,217),(209,218),(210,219)])
 

Matrix representation ►G ⊆ GL6(𝔽29)

2800000
0280000
0025000
0002500
000070
000007
,
0120000
1200000
0014900
00201500
0000149
00002015
,
1700000
0170000
00201500
0014900
00002015
0000149
,
100000
010000
0000280
0000028
001000
000100
,
100000
0280000
001000
000100
0000280
0000028

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,25,0,0,0,0,0,0,25,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,14,20,0,0,0,0,9,15,0,0,0,0,0,0,14,20,0,0,0,0,9,15],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,20,14,0,0,0,0,15,9,0,0,0,0,0,0,20,14,0,0,0,0,15,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,0,28,0,0],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;
 

64 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K···4O7A7B7C14A···14I14J···14O14P14Q14R28A···28L28M···28U
order122222244444444444···477714···1414···1414141428···2828···28
size111144282244441414141428···282222···24···48884···48···8

64 irreducible representations

dim11111111111111122222244444
type+++++++++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D7C4○D4D14D14D14D142+ (1+4)2- (1+4)D4⋊6D14D7×C4○D4D4.10D14
kernelC14.652+ (1+4)C23.11D14C22⋊Dic14C23.D14D14.D4Dic7.D4Dic7⋊3Q8C28⋊Q8D14⋊Q8C4⋊C4⋊D7C28.48D4C4×C7⋊D4C28.17D4Dic7⋊D4C7×C22.D4C22.D4Dic7C22⋊C4C4⋊C4C22×C4C2×D4C14C14C2C2C2
# reps11111211111111134963311666

In GAP, Magma, Sage, TeX 

C_{14}._{65}2_+^{(1+4)}
 
% in TeX
 
G:=Group("C14.65ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(448,1115);
 
// by ID
 
G=gap.SmallGroup(448,1115);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,675,297,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=a^7*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^7*b^-1,b*d=d*b,e*b*e=a^7*b,c*d=d*c,c*e=e*c,e*d*e=a^7*b^2*d>;
 
// generators/relations
 

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