?
G = C14.522+ (1+4) order 448 = 26·7
52nd non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.522+ (1+4), C7⋊D4⋊3Q8, C4⋊C4.97D14, C7⋊6(D4⋊3Q8), C22⋊Q8⋊13D7, C22.2(Q8×D7), (C2×Q8).75D14, D14.10(C2×Q8), D14⋊Q8⋊22C2, D14⋊3Q8⋊18C2, Dic7.Q8⋊19C2, (C2×C28).59C23, C22⋊C4.61D14, Dic7.12(C2×Q8), Dic7⋊Q8⋊16C2, Dic7⋊3Q8⋊27C2, C14.38(C22×Q8), (C2×C14).180C24, D14⋊C4.25C22, Dic7⋊4D4.3C2, (C22×C4).242D14, C2.54(D4⋊6D14), Dic7.24(C4○D4), C22⋊Dic14⋊26C2, Dic7⋊C4.30C22, C4⋊Dic7.217C22, (Q8×C14).111C22, (C2×Dic7).91C23, C23.193(C22×D7), C22.201(C23×D7), (C22×C14).208C23, (C22×C28).380C22, (C4×Dic7).109C22, (C22×D7).202C23, C23.D7.120C22, (C2×Dic14).161C22, (C22×Dic7).121C22, (D7×C4⋊C4)⋊28C2, C2.21(C2×Q8×D7), C2.51(D7×C4○D4), (C2×C14).9(C2×Q8), (C4×C7⋊D4).18C2, (C7×C22⋊Q8)⋊16C2, (C2×Dic7⋊C4)⋊41C2, C14.163(C2×C4○D4), (C2×C4×D7).100C22, (C2×C4).50(C22×D7), (C7×C4⋊C4).162C22, (C2×C7⋊D4).127C22, (C7×C22⋊C4).35C22, SmallGroup(448,1089)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 956 in 228 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×15], C22, C22 [×2], C22 [×6], C7, C2×C4 [×6], C2×C4 [×15], D4 [×4], Q8 [×4], C23, C23, D7 [×2], C14 [×3], C14 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4, C22×C4 [×5], C2×D4, C2×Q8, C2×Q8 [×2], Dic7 [×4], Dic7 [×5], C28 [×6], D14 [×2], D14 [×2], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8, C22⋊Q8 [×5], C42.C2 [×2], C4⋊Q8, Dic14 [×3], C4×D7 [×4], C2×Dic7 [×7], C2×Dic7 [×3], C7⋊D4 [×4], C2×C28 [×6], C2×C28, C7×Q8, C22×D7, C22×C14, D4⋊3Q8, C4×Dic7 [×3], Dic7⋊C4 [×11], C4⋊Dic7 [×2], D14⋊C4 [×3], C23.D7, C7×C22⋊C4 [×2], C7×C4⋊C4 [×3], C2×Dic14 [×2], C2×C4×D7 [×3], C22×Dic7 [×2], C2×C7⋊D4, C22×C28, Q8×C14, C22⋊Dic14 [×2], Dic7⋊4D4 [×2], Dic7⋊3Q8, Dic7.Q8 [×2], D7×C4⋊C4, D14⋊Q8 [×2], C2×Dic7⋊C4, C4×C7⋊D4, Dic7⋊Q8, D14⋊3Q8, C7×C22⋊Q8, C14.522+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C4○D4 [×2], C24, D14 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C22×D7 [×7], D4⋊3Q8, Q8×D7 [×2], C23×D7, D4⋊6D14, C2×Q8×D7, D7×C4○D4, C14.522+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, ab=ba, ac=ca, dad-1=eae=a-1, cbc-1=b-1, dbd-1=a7b, be=eb, cd=dc, ce=ec, ede=a7b2d >
►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 91 162 139)(2 92 163 140)(3 93 164 127)(4 94 165 128)(5 95 166 129)(6 96 167 130)(7 97 168 131)(8 98 155 132)(9 85 156 133)(10 86 157 134)(11 87 158 135)(12 88 159 136)(13 89 160 137)(14 90 161 138)(15 52 34 110)(16 53 35 111)(17 54 36 112)(18 55 37 99)(19 56 38 100)(20 43 39 101)(21 44 40 102)(22 45 41 103)(23 46 42 104)(24 47 29 105)(25 48 30 106)(26 49 31 107)(27 50 32 108)(28 51 33 109)(57 123 195 150)(58 124 196 151)(59 125 183 152)(60 126 184 153)(61 113 185 154)(62 114 186 141)(63 115 187 142)(64 116 188 143)(65 117 189 144)(66 118 190 145)(67 119 191 146)(68 120 192 147)(69 121 193 148)(70 122 194 149)(71 201 172 223)(72 202 173 224)(73 203 174 211)(74 204 175 212)(75 205 176 213)(76 206 177 214)(77 207 178 215)(78 208 179 216)(79 209 180 217)(80 210 181 218)(81 197 182 219)(82 198 169 220)(83 199 170 221)(84 200 171 222)
(1 45 8 52)(2 46 9 53)(3 47 10 54)(4 48 11 55)(5 49 12 56)(6 50 13 43)(7 51 14 44)(15 91 22 98)(16 92 23 85)(17 93 24 86)(18 94 25 87)(19 95 26 88)(20 96 27 89)(21 97 28 90)(29 134 36 127)(30 135 37 128)(31 136 38 129)(32 137 39 130)(33 138 40 131)(34 139 41 132)(35 140 42 133)(57 181 64 174)(58 182 65 175)(59 169 66 176)(60 170 67 177)(61 171 68 178)(62 172 69 179)(63 173 70 180)(71 193 78 186)(72 194 79 187)(73 195 80 188)(74 196 81 189)(75 183 82 190)(76 184 83 191)(77 185 84 192)(99 165 106 158)(100 166 107 159)(101 167 108 160)(102 168 109 161)(103 155 110 162)(104 156 111 163)(105 157 112 164)(113 200 120 207)(114 201 121 208)(115 202 122 209)(116 203 123 210)(117 204 124 197)(118 205 125 198)(119 206 126 199)(141 223 148 216)(142 224 149 217)(143 211 150 218)(144 212 151 219)(145 213 152 220)(146 214 153 221)(147 215 154 222)
(1 205 162 213)(2 204 163 212)(3 203 164 211)(4 202 165 224)(5 201 166 223)(6 200 167 222)(7 199 168 221)(8 198 155 220)(9 197 156 219)(10 210 157 218)(11 209 158 217)(12 208 159 216)(13 207 160 215)(14 206 161 214)(15 59 34 183)(16 58 35 196)(17 57 36 195)(18 70 37 194)(19 69 38 193)(20 68 39 192)(21 67 40 191)(22 66 41 190)(23 65 42 189)(24 64 29 188)(25 63 30 187)(26 62 31 186)(27 61 32 185)(28 60 33 184)(43 113 101 154)(44 126 102 153)(45 125 103 152)(46 124 104 151)(47 123 105 150)(48 122 106 149)(49 121 107 148)(50 120 108 147)(51 119 109 146)(52 118 110 145)(53 117 111 144)(54 116 112 143)(55 115 99 142)(56 114 100 141)(71 88 172 136)(72 87 173 135)(73 86 174 134)(74 85 175 133)(75 98 176 132)(76 97 177 131)(77 96 178 130)(78 95 179 129)(79 94 180 128)(80 93 181 127)(81 92 182 140)(82 91 169 139)(83 90 170 138)(84 89 171 137)
(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(29 39)(30 38)(31 37)(32 36)(33 35)(40 42)(43 47)(44 46)(48 56)(49 55)(50 54)(51 53)(57 192)(58 191)(59 190)(60 189)(61 188)(62 187)(63 186)(64 185)(65 184)(66 183)(67 196)(68 195)(69 194)(70 193)(71 173)(72 172)(73 171)(74 170)(75 169)(76 182)(77 181)(78 180)(79 179)(80 178)(81 177)(82 176)(83 175)(84 174)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(99 107)(100 106)(101 105)(102 104)(108 112)(109 111)(113 143)(114 142)(115 141)(116 154)(117 153)(118 152)(119 151)(120 150)(121 149)(122 148)(123 147)(124 146)(125 145)(126 144)(127 137)(128 136)(129 135)(130 134)(131 133)(138 140)(156 168)(157 167)(158 166)(159 165)(160 164)(161 163)(197 214)(198 213)(199 212)(200 211)(201 224)(202 223)(203 222)(204 221)(205 220)(206 219)(207 218)(208 217)(209 216)(210 215)
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), (1,91,162,139)(2,92,163,140)(3,93,164,127)(4,94,165,128)(5,95,166,129)(6,96,167,130)(7,97,168,131)(8,98,155,132)(9,85,156,133)(10,86,157,134)(11,87,158,135)(12,88,159,136)(13,89,160,137)(14,90,161,138)(15,52,34,110)(16,53,35,111)(17,54,36,112)(18,55,37,99)(19,56,38,100)(20,43,39,101)(21,44,40,102)(22,45,41,103)(23,46,42,104)(24,47,29,105)(25,48,30,106)(26,49,31,107)(27,50,32,108)(28,51,33,109)(57,123,195,150)(58,124,196,151)(59,125,183,152)(60,126,184,153)(61,113,185,154)(62,114,186,141)(63,115,187,142)(64,116,188,143)(65,117,189,144)(66,118,190,145)(67,119,191,146)(68,120,192,147)(69,121,193,148)(70,122,194,149)(71,201,172,223)(72,202,173,224)(73,203,174,211)(74,204,175,212)(75,205,176,213)(76,206,177,214)(77,207,178,215)(78,208,179,216)(79,209,180,217)(80,210,181,218)(81,197,182,219)(82,198,169,220)(83,199,170,221)(84,200,171,222), (1,45,8,52)(2,46,9,53)(3,47,10,54)(4,48,11,55)(5,49,12,56)(6,50,13,43)(7,51,14,44)(15,91,22,98)(16,92,23,85)(17,93,24,86)(18,94,25,87)(19,95,26,88)(20,96,27,89)(21,97,28,90)(29,134,36,127)(30,135,37,128)(31,136,38,129)(32,137,39,130)(33,138,40,131)(34,139,41,132)(35,140,42,133)(57,181,64,174)(58,182,65,175)(59,169,66,176)(60,170,67,177)(61,171,68,178)(62,172,69,179)(63,173,70,180)(71,193,78,186)(72,194,79,187)(73,195,80,188)(74,196,81,189)(75,183,82,190)(76,184,83,191)(77,185,84,192)(99,165,106,158)(100,166,107,159)(101,167,108,160)(102,168,109,161)(103,155,110,162)(104,156,111,163)(105,157,112,164)(113,200,120,207)(114,201,121,208)(115,202,122,209)(116,203,123,210)(117,204,124,197)(118,205,125,198)(119,206,126,199)(141,223,148,216)(142,224,149,217)(143,211,150,218)(144,212,151,219)(145,213,152,220)(146,214,153,221)(147,215,154,222), (1,205,162,213)(2,204,163,212)(3,203,164,211)(4,202,165,224)(5,201,166,223)(6,200,167,222)(7,199,168,221)(8,198,155,220)(9,197,156,219)(10,210,157,218)(11,209,158,217)(12,208,159,216)(13,207,160,215)(14,206,161,214)(15,59,34,183)(16,58,35,196)(17,57,36,195)(18,70,37,194)(19,69,38,193)(20,68,39,192)(21,67,40,191)(22,66,41,190)(23,65,42,189)(24,64,29,188)(25,63,30,187)(26,62,31,186)(27,61,32,185)(28,60,33,184)(43,113,101,154)(44,126,102,153)(45,125,103,152)(46,124,104,151)(47,123,105,150)(48,122,106,149)(49,121,107,148)(50,120,108,147)(51,119,109,146)(52,118,110,145)(53,117,111,144)(54,116,112,143)(55,115,99,142)(56,114,100,141)(71,88,172,136)(72,87,173,135)(73,86,174,134)(74,85,175,133)(75,98,176,132)(76,97,177,131)(77,96,178,130)(78,95,179,129)(79,94,180,128)(80,93,181,127)(81,92,182,140)(82,91,169,139)(83,90,170,138)(84,89,171,137), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(29,39)(30,38)(31,37)(32,36)(33,35)(40,42)(43,47)(44,46)(48,56)(49,55)(50,54)(51,53)(57,192)(58,191)(59,190)(60,189)(61,188)(62,187)(63,186)(64,185)(65,184)(66,183)(67,196)(68,195)(69,194)(70,193)(71,173)(72,172)(73,171)(74,170)(75,169)(76,182)(77,181)(78,180)(79,179)(80,178)(81,177)(82,176)(83,175)(84,174)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(99,107)(100,106)(101,105)(102,104)(108,112)(109,111)(113,143)(114,142)(115,141)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,137)(128,136)(129,135)(130,134)(131,133)(138,140)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(197,214)(198,213)(199,212)(200,211)(201,224)(202,223)(203,222)(204,221)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)>;
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), (1,91,162,139)(2,92,163,140)(3,93,164,127)(4,94,165,128)(5,95,166,129)(6,96,167,130)(7,97,168,131)(8,98,155,132)(9,85,156,133)(10,86,157,134)(11,87,158,135)(12,88,159,136)(13,89,160,137)(14,90,161,138)(15,52,34,110)(16,53,35,111)(17,54,36,112)(18,55,37,99)(19,56,38,100)(20,43,39,101)(21,44,40,102)(22,45,41,103)(23,46,42,104)(24,47,29,105)(25,48,30,106)(26,49,31,107)(27,50,32,108)(28,51,33,109)(57,123,195,150)(58,124,196,151)(59,125,183,152)(60,126,184,153)(61,113,185,154)(62,114,186,141)(63,115,187,142)(64,116,188,143)(65,117,189,144)(66,118,190,145)(67,119,191,146)(68,120,192,147)(69,121,193,148)(70,122,194,149)(71,201,172,223)(72,202,173,224)(73,203,174,211)(74,204,175,212)(75,205,176,213)(76,206,177,214)(77,207,178,215)(78,208,179,216)(79,209,180,217)(80,210,181,218)(81,197,182,219)(82,198,169,220)(83,199,170,221)(84,200,171,222), (1,45,8,52)(2,46,9,53)(3,47,10,54)(4,48,11,55)(5,49,12,56)(6,50,13,43)(7,51,14,44)(15,91,22,98)(16,92,23,85)(17,93,24,86)(18,94,25,87)(19,95,26,88)(20,96,27,89)(21,97,28,90)(29,134,36,127)(30,135,37,128)(31,136,38,129)(32,137,39,130)(33,138,40,131)(34,139,41,132)(35,140,42,133)(57,181,64,174)(58,182,65,175)(59,169,66,176)(60,170,67,177)(61,171,68,178)(62,172,69,179)(63,173,70,180)(71,193,78,186)(72,194,79,187)(73,195,80,188)(74,196,81,189)(75,183,82,190)(76,184,83,191)(77,185,84,192)(99,165,106,158)(100,166,107,159)(101,167,108,160)(102,168,109,161)(103,155,110,162)(104,156,111,163)(105,157,112,164)(113,200,120,207)(114,201,121,208)(115,202,122,209)(116,203,123,210)(117,204,124,197)(118,205,125,198)(119,206,126,199)(141,223,148,216)(142,224,149,217)(143,211,150,218)(144,212,151,219)(145,213,152,220)(146,214,153,221)(147,215,154,222), (1,205,162,213)(2,204,163,212)(3,203,164,211)(4,202,165,224)(5,201,166,223)(6,200,167,222)(7,199,168,221)(8,198,155,220)(9,197,156,219)(10,210,157,218)(11,209,158,217)(12,208,159,216)(13,207,160,215)(14,206,161,214)(15,59,34,183)(16,58,35,196)(17,57,36,195)(18,70,37,194)(19,69,38,193)(20,68,39,192)(21,67,40,191)(22,66,41,190)(23,65,42,189)(24,64,29,188)(25,63,30,187)(26,62,31,186)(27,61,32,185)(28,60,33,184)(43,113,101,154)(44,126,102,153)(45,125,103,152)(46,124,104,151)(47,123,105,150)(48,122,106,149)(49,121,107,148)(50,120,108,147)(51,119,109,146)(52,118,110,145)(53,117,111,144)(54,116,112,143)(55,115,99,142)(56,114,100,141)(71,88,172,136)(72,87,173,135)(73,86,174,134)(74,85,175,133)(75,98,176,132)(76,97,177,131)(77,96,178,130)(78,95,179,129)(79,94,180,128)(80,93,181,127)(81,92,182,140)(82,91,169,139)(83,90,170,138)(84,89,171,137), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(29,39)(30,38)(31,37)(32,36)(33,35)(40,42)(43,47)(44,46)(48,56)(49,55)(50,54)(51,53)(57,192)(58,191)(59,190)(60,189)(61,188)(62,187)(63,186)(64,185)(65,184)(66,183)(67,196)(68,195)(69,194)(70,193)(71,173)(72,172)(73,171)(74,170)(75,169)(76,182)(77,181)(78,180)(79,179)(80,178)(81,177)(82,176)(83,175)(84,174)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(99,107)(100,106)(101,105)(102,104)(108,112)(109,111)(113,143)(114,142)(115,141)(116,154)(117,153)(118,152)(119,151)(120,150)(121,149)(122,148)(123,147)(124,146)(125,145)(126,144)(127,137)(128,136)(129,135)(130,134)(131,133)(138,140)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(197,214)(198,213)(199,212)(200,211)(201,224)(202,223)(203,222)(204,221)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215) );
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)], [(1,91,162,139),(2,92,163,140),(3,93,164,127),(4,94,165,128),(5,95,166,129),(6,96,167,130),(7,97,168,131),(8,98,155,132),(9,85,156,133),(10,86,157,134),(11,87,158,135),(12,88,159,136),(13,89,160,137),(14,90,161,138),(15,52,34,110),(16,53,35,111),(17,54,36,112),(18,55,37,99),(19,56,38,100),(20,43,39,101),(21,44,40,102),(22,45,41,103),(23,46,42,104),(24,47,29,105),(25,48,30,106),(26,49,31,107),(27,50,32,108),(28,51,33,109),(57,123,195,150),(58,124,196,151),(59,125,183,152),(60,126,184,153),(61,113,185,154),(62,114,186,141),(63,115,187,142),(64,116,188,143),(65,117,189,144),(66,118,190,145),(67,119,191,146),(68,120,192,147),(69,121,193,148),(70,122,194,149),(71,201,172,223),(72,202,173,224),(73,203,174,211),(74,204,175,212),(75,205,176,213),(76,206,177,214),(77,207,178,215),(78,208,179,216),(79,209,180,217),(80,210,181,218),(81,197,182,219),(82,198,169,220),(83,199,170,221),(84,200,171,222)], [(1,45,8,52),(2,46,9,53),(3,47,10,54),(4,48,11,55),(5,49,12,56),(6,50,13,43),(7,51,14,44),(15,91,22,98),(16,92,23,85),(17,93,24,86),(18,94,25,87),(19,95,26,88),(20,96,27,89),(21,97,28,90),(29,134,36,127),(30,135,37,128),(31,136,38,129),(32,137,39,130),(33,138,40,131),(34,139,41,132),(35,140,42,133),(57,181,64,174),(58,182,65,175),(59,169,66,176),(60,170,67,177),(61,171,68,178),(62,172,69,179),(63,173,70,180),(71,193,78,186),(72,194,79,187),(73,195,80,188),(74,196,81,189),(75,183,82,190),(76,184,83,191),(77,185,84,192),(99,165,106,158),(100,166,107,159),(101,167,108,160),(102,168,109,161),(103,155,110,162),(104,156,111,163),(105,157,112,164),(113,200,120,207),(114,201,121,208),(115,202,122,209),(116,203,123,210),(117,204,124,197),(118,205,125,198),(119,206,126,199),(141,223,148,216),(142,224,149,217),(143,211,150,218),(144,212,151,219),(145,213,152,220),(146,214,153,221),(147,215,154,222)], [(1,205,162,213),(2,204,163,212),(3,203,164,211),(4,202,165,224),(5,201,166,223),(6,200,167,222),(7,199,168,221),(8,198,155,220),(9,197,156,219),(10,210,157,218),(11,209,158,217),(12,208,159,216),(13,207,160,215),(14,206,161,214),(15,59,34,183),(16,58,35,196),(17,57,36,195),(18,70,37,194),(19,69,38,193),(20,68,39,192),(21,67,40,191),(22,66,41,190),(23,65,42,189),(24,64,29,188),(25,63,30,187),(26,62,31,186),(27,61,32,185),(28,60,33,184),(43,113,101,154),(44,126,102,153),(45,125,103,152),(46,124,104,151),(47,123,105,150),(48,122,106,149),(49,121,107,148),(50,120,108,147),(51,119,109,146),(52,118,110,145),(53,117,111,144),(54,116,112,143),(55,115,99,142),(56,114,100,141),(71,88,172,136),(72,87,173,135),(73,86,174,134),(74,85,175,133),(75,98,176,132),(76,97,177,131),(77,96,178,130),(78,95,179,129),(79,94,180,128),(80,93,181,127),(81,92,182,140),(82,91,169,139),(83,90,170,138),(84,89,171,137)], [(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(29,39),(30,38),(31,37),(32,36),(33,35),(40,42),(43,47),(44,46),(48,56),(49,55),(50,54),(51,53),(57,192),(58,191),(59,190),(60,189),(61,188),(62,187),(63,186),(64,185),(65,184),(66,183),(67,196),(68,195),(69,194),(70,193),(71,173),(72,172),(73,171),(74,170),(75,169),(76,182),(77,181),(78,180),(79,179),(80,178),(81,177),(82,176),(83,175),(84,174),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(99,107),(100,106),(101,105),(102,104),(108,112),(109,111),(113,143),(114,142),(115,141),(116,154),(117,153),(118,152),(119,151),(120,150),(121,149),(122,148),(123,147),(124,146),(125,145),(126,144),(127,137),(128,136),(129,135),(130,134),(131,133),(138,140),(156,168),(157,167),(158,166),(159,165),(160,164),(161,163),(197,214),(198,213),(199,212),(200,211),(201,224),(202,223),(203,222),(204,221),(205,220),(206,219),(207,218),(208,217),(209,216),(210,215)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 21 | 0 | 0 |
| 0 | 0 | 8 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 23 | 8 | 0 | 0 | 0 | 0 |
| 28 | 6 | 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 | 22 | 1 |
| 7 | 15 | 0 | 0 | 0 | 0 |
| 16 | 22 | 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 | 12 |
| 7 | 15 | 0 | 0 | 0 | 0 |
| 16 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 3 | 0 | 0 |
| 0 | 0 | 8 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 7 |
| 0 | 0 | 0 | 0 | 19 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 26 | 0 | 0 |
| 0 | 0 | 21 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 7 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,26,8,0,0,0,0,21,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[23,28,0,0,0,0,8,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,22,0,0,0,0,0,1],[7,16,0,0,0,0,15,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[7,16,0,0,0,0,15,22,0,0,0,0,0,0,8,8,0,0,0,0,3,21,0,0,0,0,0,0,19,19,0,0,0,0,7,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,21,0,0,0,0,26,8,0,0,0,0,0,0,1,7,0,0,0,0,0,28] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 2 | 2 | 4 | ··· | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | Q8×D7 | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.522+ (1+4) | C22⋊Dic14 | Dic7⋊4D4 | Dic7⋊3Q8 | Dic7.Q8 | D7×C4⋊C4 | D14⋊Q8 | C2×Dic7⋊C4 | C4×C7⋊D4 | Dic7⋊Q8 | D14⋊3Q8 | C7×C22⋊Q8 | C7⋊D4 | C22⋊Q8 | Dic7 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{52}2_+^{(1+4)} % in TeX
G:=Group("C14.52ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1089);
// by ID
G=gap.SmallGroup(448,1089);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,100,570,409,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^7*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a^7*b^2*d>;
// generators/relations
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