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G = C14.822- (1+4)  order 448 = 26·7

37th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C14.822- (1+4), C7⋊D4⋊7D4, C28⋊Q8⋊32C2, C7⋊6(D4⋊6D4), C4⋊C4.106D14, D14.23(C2×D4), D14⋊D4⋊30C2, C22.13(D4×D7), D14⋊Q8⋊28C2, Dic7⋊6(C4○D4), (C2×D4).164D14, (C2×C28).72C23, C22⋊C4.27D14, Dic7.27(C2×D4), C14.84(C22×D4), C22.D4⋊5D7, Dic7⋊D4⋊21C2, Dic7⋊4D4⋊19C2, D14.D4⋊30C2, D14.5D4⋊28C2, (C2×C14).199C24, D14⋊C4.32C22, (C22×C4).257D14, C22⋊Dic14⋊31C2, (C2×D28).222C22, (D4×C14).137C22, C22.D28⋊20C2, Dic7⋊C4.41C22, C4⋊Dic7.227C22, (C22×C14).34C23, C22.220(C23×D7), C23.201(C22×D7), C23.D7.42C22, (C22×C28).113C22, (C2×Dic7).103C23, (C4×Dic7).123C22, (C22×D7).207C23, C2.43(D4.10D14), (C2×Dic14).168C22, (C22×Dic7).129C22, C2.57(C2×D4×D7), (D7×C4⋊C4)⋊32C2, C2.61(D7×C4○D4), (C2×C4○D28)⋊12C2, (C2×C14).60(C2×D4), (C2×D4⋊2D7)⋊16C2, (C2×Dic7⋊C4)⋊27C2, C14.173(C2×C4○D4), (C2×C4×D7).110C22, (C2×C4).62(C22×D7), (C7×C4⋊C4).176C22, (C7×C22.D4)⋊7C2, (C2×C7⋊D4).134C22, (C7×C22⋊C4).51C22, SmallGroup(448,1108)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C14.822- (1+4)
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C7⋊D4 — C2×C4○D28 — C14.822- (1+4)
Lower central
C7 — C2×C14 — C14.822- (1+4)

Subgroups: 1356 in 292 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×13], C22, C22 [×2], C22 [×12], C7, C2×C4 [×5], C2×C4 [×22], D4 [×14], Q8 [×4], C23 [×2], C23 [×2], D7 [×3], C14 [×3], C14 [×3], C42, C22⋊C4 [×3], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×7], C2×D4, C2×D4 [×5], C2×Q8 [×2], C4○D4 [×8], Dic7 [×4], Dic7 [×4], C28 [×5], D14 [×2], D14 [×5], C2×C14, C2×C14 [×2], C2×C14 [×5], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C22.D4 [×3], C4⋊Q8, C2×C4○D4 [×2], Dic14 [×4], C4×D7 [×8], D28 [×2], C2×Dic7 [×6], C2×Dic7 [×6], C7⋊D4 [×4], C7⋊D4 [×6], C2×C28 [×5], C2×C28 [×2], C7×D4 [×2], C22×D7 [×2], C22×C14 [×2], D4⋊6D4, C4×Dic7, Dic7⋊C4 [×6], C4⋊Dic7 [×2], D14⋊C4 [×4], C23.D7, C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×Dic14 [×2], C2×C4×D7 [×4], C2×D28, C4○D28 [×4], D4⋊2D7 [×4], C22×Dic7 [×3], C2×C7⋊D4 [×4], C22×C28, D4×C14, C22⋊Dic14, Dic7⋊4D4 [×2], D14.D4, D14⋊D4, C22.D28, C28⋊Q8, D7×C4⋊C4, D14.5D4, D14⋊Q8, C2×Dic7⋊C4, Dic7⋊D4, C7×C22.D4, C2×C4○D28, C2×D4⋊2D7, C14.822- (1+4)

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2- (1+4), C22×D7 [×7], D4⋊6D4, D4×D7 [×2], C23×D7, C2×D4×D7, D7×C4○D4, D4.10D14, C14.822- (1+4)

Generators and relations
 G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=b2, bab-1=cac=dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=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 83 221 116)(2 82 222 115)(3 81 223 114)(4 80 224 113)(5 79 211 126)(6 78 212 125)(7 77 213 124)(8 76 214 123)(9 75 215 122)(10 74 216 121)(11 73 217 120)(12 72 218 119)(13 71 219 118)(14 84 220 117)(15 31 97 196)(16 30 98 195)(17 29 85 194)(18 42 86 193)(19 41 87 192)(20 40 88 191)(21 39 89 190)(22 38 90 189)(23 37 91 188)(24 36 92 187)(25 35 93 186)(26 34 94 185)(27 33 95 184)(28 32 96 183)(43 206 129 65)(44 205 130 64)(45 204 131 63)(46 203 132 62)(47 202 133 61)(48 201 134 60)(49 200 135 59)(50 199 136 58)(51 198 137 57)(52 197 138 70)(53 210 139 69)(54 209 140 68)(55 208 127 67)(56 207 128 66)(99 171 146 161)(100 170 147 160)(101 169 148 159)(102 182 149 158)(103 181 150 157)(104 180 151 156)(105 179 152 155)(106 178 153 168)(107 177 154 167)(108 176 141 166)(109 175 142 165)(110 174 143 164)(111 173 144 163)(112 172 145 162)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽29)

2800000
0280000
0002100
00111800
0000280
0000028
,
320000
24260000
0025100
0014400
000010
000001
,
2800000
310000
0025100
0014400
0000280
0000028
,
1700000
0170000
0025100
0014400
0000118
0000028
,
1200000
0120000
0028000
0002800
0000922
00002820

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,11,0,0,0,0,21,18,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[3,24,0,0,0,0,2,26,0,0,0,0,0,0,25,14,0,0,0,0,1,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,3,0,0,0,0,0,1,0,0,0,0,0,0,25,14,0,0,0,0,1,4,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,25,14,0,0,0,0,1,4,0,0,0,0,0,0,1,0,0,0,0,0,18,28],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,9,28,0,0,0,0,22,20] >;
 

67 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G···4L4M4N4O7A7B7C14A···14I14J···14O14P14Q14R28A···28L28M···28U
order12222222224444444···444477714···1414···1414141428···2828···28
size111122414142822444414···142828282222···24···48884···48···8

67 irreducible representations

dim11111111111111122222224444
type+++++++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D7C4○D4D14D14D14D142- (1+4)D4×D7D7×C4○D4D4.10D14
kernelC14.822- (1+4)C22⋊Dic14Dic7⋊4D4D14.D4D14⋊D4C22.D28C28⋊Q8D7×C4⋊C4D14.5D4D14⋊Q8C2×Dic7⋊C4Dic7⋊D4C7×C22.D4C2×C4○D28C2×D4⋊2D7C7⋊D4C22.D4Dic7C22⋊C4C4⋊C4C22×C4C2×D4C14C22C2C2
# reps11211111111111143496331666

In GAP, Magma, Sage, TeX 

C_{14}._{82}2_-^{(1+4)}
 
% in TeX
 
G:=Group("C14.82ES-(2,2)");
 
// GroupNames label
 
G:=SmallGroup(448,1108);
 
// by ID
 
G=gap.SmallGroup(448,1108);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,184,570,185,136,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=b^2,b*a*b^-1=c*a*c=d*a*d^-1=a^-1,a*e=e*a,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=a^7*b^2*d>;
 
// generators/relations
 

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