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G = C14.1442+ (1+4)  order 448 = 26·7

53rd non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C14.1442+ (1+4), C14.1062- (1+4), C4○D4⋊5Dic7, D4⋊8(C2×Dic7), Q8⋊7(C2×Dic7), (Q8×Dic7)⋊28C2, (D4×Dic7)⋊41C2, (C2×D4).253D14, C14.51(C23×C4), (C2×Q8).209D14, C2.5(D4⋊8D14), C28.100(C22×C4), (C2×C14).313C24, (C2×C28).560C23, (C22×C4).286D14, C2.13(C23×Dic7), C4.22(C22×Dic7), C22.49(C23×D7), (D4×C14).275C22, C4⋊Dic7.392C22, (Q8×C14).242C22, C23.210(C22×D7), C2.5(D4.10D14), C23.21D14⋊37C2, C22.4(C22×Dic7), (C22×C14).239C23, (C22×C28).295C22, C7⋊5(C23.33C23), (C2×Dic7).292C23, (C4×Dic7).175C22, C23.D7.135C22, (C22×Dic7).167C22, (C7×C4○D4)⋊6C4, (C2×C28)⋊17(C2×C4), (C7×D4)⋊22(C2×C4), (C7×Q8)⋊20(C2×C4), (C2×C4)⋊5(C2×Dic7), (C2×C4⋊Dic7)⋊47C2, (C2×C4○D4).12D7, (C14×C4○D4).14C2, (C2×C14).31(C22×C4), (C2×C4).638(C22×D7), SmallGroup(448,1280)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C14.1442+ (1+4)
Chief series
C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — D4×Dic7 — C14.1442+ (1+4)
Lower central
C7 — C14 — C14.1442+ (1+4)

Subgroups: 916 in 294 conjugacy classes, 191 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×8], C22, C22 [×6], C22 [×6], C7, C2×C4, C2×C4 [×15], C2×C4 [×14], D4 [×12], Q8 [×4], C23 [×3], C14 [×3], C14 [×6], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic7 [×8], C28 [×8], C2×C14, C2×C14 [×6], C2×C14 [×6], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C2×Dic7 [×8], C2×Dic7 [×6], C2×C28, C2×C28 [×15], C7×D4 [×12], C7×Q8 [×4], C22×C14 [×3], C23.33C23, C4×Dic7 [×6], C4⋊Dic7, C4⋊Dic7 [×9], C23.D7 [×6], C22×Dic7 [×6], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×8], C2×C4⋊Dic7 [×3], C23.21D14 [×3], D4×Dic7 [×6], Q8×Dic7 [×2], C14×C4○D4, C14.1442+ (1+4)

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, C22×C4 [×14], C24, Dic7 [×8], D14 [×7], C23×C4, 2+ (1+4), 2- (1+4), C2×Dic7 [×28], C22×D7 [×7], C23.33C23, C22×Dic7 [×14], C23×D7, D4⋊8D14, D4.10D14, C23×Dic7, C14.1442+ (1+4)

Generators and relations
 G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

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)

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

(43 162)(44 163)(45 164)(46 165)(47 166)(48 167)(49 168)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(71 119)(72 120)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(127 171)(128 172)(129 173)(130 174)(131 175)(132 176)(133 177)(134 178)(135 179)(136 180)(137 181)(138 182)(139 169)(140 170)(197 213)(198 214)(199 215)(200 216)(201 217)(202 218)(203 219)(204 220)(205 221)(206 222)(207 223)(208 224)(209 211)(210 212)

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

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

Matrix representation ►G ⊆ GL6(𝔽29)

440000
25180000
008800
0021300
000098
0000132
,
2800000
0280000
00218821
0011271714
0000311
0000726
,
2800000
0280000
001000
000100
00014280
00154028
,
100000
010000
00271100
0018200
0000311
0000726
,
2110000
18270000
0051600
0022400
002817212
002517118

G:=sub<GL(6,GF(29))| [4,25,0,0,0,0,4,18,0,0,0,0,0,0,8,21,0,0,0,0,8,3,0,0,0,0,0,0,9,13,0,0,0,0,8,2],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,2,11,0,0,0,0,18,27,0,0,0,0,8,17,3,7,0,0,21,14,11,26],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,15,0,0,0,1,14,4,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,18,0,0,0,0,11,2,0,0,0,0,0,0,3,7,0,0,0,0,11,26],[2,18,0,0,0,0,11,27,0,0,0,0,0,0,5,2,28,25,0,0,16,24,17,17,0,0,0,0,21,11,0,0,0,0,2,8] >;
 

94 conjugacy classes

class 1 2A2B2C2D···2I4A···4H4I···4X7A7B7C14A···14I14J···14AA28A···28L28M···28AD
order12222···24···44···477714···1414···1428···2828···28
size11112···22···214···142222···24···42···24···4

94 irreducible representations

dim1111111222224444
type++++++++++-+-+-
imageC1C2C2C2C2C2C4D7D14D14D14Dic72+ (1+4)2- (1+4)D4⋊8D14D4.10D14
kernelC14.1442+ (1+4)C2×C4⋊Dic7C23.21D14D4×Dic7Q8×Dic7C14×C4○D4C7×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C14C14C2C2
# reps133621163993241166

In GAP, Magma, Sage, TeX 

C_{14}._{144}2_+^{(1+4)}
 
% in TeX
 
G:=Group("C14.144ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(448,1280);
 
// by ID
 
G=gap.SmallGroup(448,1280);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,387,1123,136,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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