direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×D4×Q8, C14.1192- (1+4), C4⋊2(Q8×C14), C4⋊Q8⋊14C14, C28⋊10(C2×Q8), (C4×Q8)⋊12C14, (Q8×C28)⋊32C2, C4.43(D4×C14), C22⋊2(Q8×C14), (C4×D4).10C14, (D4×C28).25C2, C28.404(C2×D4), C22⋊Q8⋊14C14, (C22×Q8)⋊7C14, C42.44(C2×C14), C14.63(C22×Q8), (C2×C14).369C24, (C4×C28).285C22, (C2×C28).676C23, C14.197(C22×D4), (D4×C14).334C22, C22.43(C23×C14), C23.43(C22×C14), (Q8×C14).275C22, C2.11(C7×2- (1+4)), (C22×C14).265C23, (C22×C28).455C22, C2.9(Q8×C2×C14), (C7×C4⋊Q8)⋊35C2, (Q8×C2×C14)⋊19C2, (C2×C14)⋊8(C2×Q8), C2.21(D4×C2×C14), C4⋊C4.32(C2×C14), (C7×C22⋊Q8)⋊41C2, (C2×D4).80(C2×C14), (C2×Q8).62(C2×C14), (C7×C4⋊C4).397C22, C22⋊C4.20(C2×C14), (C22×C4).67(C2×C14), (C2×C4).34(C22×C14), (C7×C22⋊C4).153C22, SmallGroup(448,1332)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 378 in 280 conjugacy classes, 182 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×9], C22, C22 [×4], C22 [×4], C7, C2×C4, C2×C4 [×12], C2×C4 [×12], D4 [×4], Q8 [×4], Q8 [×12], C23 [×2], C14 [×3], C14 [×4], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×8], C28 [×8], C28 [×9], C2×C14, C2×C14 [×4], C2×C14 [×4], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], C2×C28, C2×C28 [×12], C2×C28 [×12], C7×D4 [×4], C7×Q8 [×4], C7×Q8 [×12], C22×C14 [×2], D4×Q8, C4×C28 [×3], C7×C22⋊C4 [×6], C7×C4⋊C4 [×12], C22×C28 [×6], D4×C14, Q8×C14, Q8×C14 [×6], Q8×C14 [×8], D4×C28 [×3], Q8×C28, C7×C22⋊Q8 [×6], C7×C4⋊Q8 [×3], Q8×C2×C14 [×2], C7×D4×Q8
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], Q8 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C2×C14 [×35], C22×D4, C22×Q8, 2- (1+4), C7×D4 [×4], C7×Q8 [×4], C22×C14 [×15], D4×Q8, D4×C14 [×6], Q8×C14 [×6], C23×C14, D4×C2×C14, Q8×C2×C14, C7×2- (1+4), C7×D4×Q8
Generators and relations
G = < a,b,c,d,e | a7=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►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 94 35 91)(2 95 29 85)(3 96 30 86)(4 97 31 87)(5 98 32 88)(6 92 33 89)(7 93 34 90)(8 177 18 174)(9 178 19 175)(10 179 20 169)(11 180 21 170)(12 181 15 171)(13 182 16 172)(14 176 17 173)(22 167 223 185)(23 168 224 186)(24 162 218 187)(25 163 219 188)(26 164 220 189)(27 165 221 183)(28 166 222 184)(36 82 44 100)(37 83 45 101)(38 84 46 102)(39 78 47 103)(40 79 48 104)(41 80 49 105)(42 81 43 99)(50 106 75 131)(51 107 76 132)(52 108 77 133)(53 109 71 127)(54 110 72 128)(55 111 73 129)(56 112 74 130)(57 113 67 123)(58 114 68 124)(59 115 69 125)(60 116 70 126)(61 117 64 120)(62 118 65 121)(63 119 66 122)(134 190 159 215)(135 191 160 216)(136 192 161 217)(137 193 155 211)(138 194 156 212)(139 195 157 213)(140 196 158 214)(141 197 151 207)(142 198 152 208)(143 199 153 209)(144 200 154 210)(145 201 148 204)(146 202 149 205)(147 203 150 206)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 224)(9 218)(10 219)(11 220)(12 221)(13 222)(14 223)(15 27)(16 28)(17 22)(18 23)(19 24)(20 25)(21 26)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(92 101)(93 102)(94 103)(95 104)(96 105)(97 99)(98 100)(106 119)(107 113)(108 114)(109 115)(110 116)(111 117)(112 118)(120 129)(121 130)(122 131)(123 132)(124 133)(125 127)(126 128)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 160)(142 161)(143 155)(144 156)(145 157)(146 158)(147 159)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 188)(170 189)(171 183)(172 184)(173 185)(174 186)(175 187)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)(204 213)(205 214)(206 215)(207 216)(208 217)(209 211)(210 212)
(1 50 39 66)(2 51 40 67)(3 52 41 68)(4 53 42 69)(5 54 36 70)(6 55 37 64)(7 56 38 65)(8 214 23 202)(9 215 24 203)(10 216 25 197)(11 217 26 198)(12 211 27 199)(13 212 28 200)(14 213 22 201)(15 193 221 209)(16 194 222 210)(17 195 223 204)(18 196 224 205)(19 190 218 206)(20 191 219 207)(21 192 220 208)(29 76 48 57)(30 77 49 58)(31 71 43 59)(32 72 44 60)(33 73 45 61)(34 74 46 62)(35 75 47 63)(78 122 94 106)(79 123 95 107)(80 124 96 108)(81 125 97 109)(82 126 98 110)(83 120 92 111)(84 121 93 112)(85 132 104 113)(86 133 105 114)(87 127 99 115)(88 128 100 116)(89 129 101 117)(90 130 102 118)(91 131 103 119)(134 162 150 178)(135 163 151 179)(136 164 152 180)(137 165 153 181)(138 166 154 182)(139 167 148 176)(140 168 149 177)(141 169 160 188)(142 170 161 189)(143 171 155 183)(144 172 156 184)(145 173 157 185)(146 174 158 186)(147 175 159 187)
(1 134 39 150)(2 135 40 151)(3 136 41 152)(4 137 42 153)(5 138 36 154)(6 139 37 148)(7 140 38 149)(8 118 23 130)(9 119 24 131)(10 113 25 132)(11 114 26 133)(12 115 27 127)(13 116 28 128)(14 117 22 129)(15 125 221 109)(16 126 222 110)(17 120 223 111)(18 121 224 112)(19 122 218 106)(20 123 219 107)(21 124 220 108)(29 160 48 141)(30 161 49 142)(31 155 43 143)(32 156 44 144)(33 157 45 145)(34 158 46 146)(35 159 47 147)(50 178 66 162)(51 179 67 163)(52 180 68 164)(53 181 69 165)(54 182 70 166)(55 176 64 167)(56 177 65 168)(57 188 76 169)(58 189 77 170)(59 183 71 171)(60 184 72 172)(61 185 73 173)(62 186 74 174)(63 187 75 175)(78 206 94 190)(79 207 95 191)(80 208 96 192)(81 209 97 193)(82 210 98 194)(83 204 92 195)(84 205 93 196)(85 216 104 197)(86 217 105 198)(87 211 99 199)(88 212 100 200)(89 213 101 201)(90 214 102 202)(91 215 103 203)
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), (1,94,35,91)(2,95,29,85)(3,96,30,86)(4,97,31,87)(5,98,32,88)(6,92,33,89)(7,93,34,90)(8,177,18,174)(9,178,19,175)(10,179,20,169)(11,180,21,170)(12,181,15,171)(13,182,16,172)(14,176,17,173)(22,167,223,185)(23,168,224,186)(24,162,218,187)(25,163,219,188)(26,164,220,189)(27,165,221,183)(28,166,222,184)(36,82,44,100)(37,83,45,101)(38,84,46,102)(39,78,47,103)(40,79,48,104)(41,80,49,105)(42,81,43,99)(50,106,75,131)(51,107,76,132)(52,108,77,133)(53,109,71,127)(54,110,72,128)(55,111,73,129)(56,112,74,130)(57,113,67,123)(58,114,68,124)(59,115,69,125)(60,116,70,126)(61,117,64,120)(62,118,65,121)(63,119,66,122)(134,190,159,215)(135,191,160,216)(136,192,161,217)(137,193,155,211)(138,194,156,212)(139,195,157,213)(140,196,158,214)(141,197,151,207)(142,198,152,208)(143,199,153,209)(144,200,154,210)(145,201,148,204)(146,202,149,205)(147,203,150,206), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,224)(9,218)(10,219)(11,220)(12,221)(13,222)(14,223)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,101)(93,102)(94,103)(95,104)(96,105)(97,99)(98,100)(106,119)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118)(120,129)(121,130)(122,131)(123,132)(124,133)(125,127)(126,128)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,160)(142,161)(143,155)(144,156)(145,157)(146,158)(147,159)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,188)(170,189)(171,183)(172,184)(173,185)(174,186)(175,187)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,213)(205,214)(206,215)(207,216)(208,217)(209,211)(210,212), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,214,23,202)(9,215,24,203)(10,216,25,197)(11,217,26,198)(12,211,27,199)(13,212,28,200)(14,213,22,201)(15,193,221,209)(16,194,222,210)(17,195,223,204)(18,196,224,205)(19,190,218,206)(20,191,219,207)(21,192,220,208)(29,76,48,57)(30,77,49,58)(31,71,43,59)(32,72,44,60)(33,73,45,61)(34,74,46,62)(35,75,47,63)(78,122,94,106)(79,123,95,107)(80,124,96,108)(81,125,97,109)(82,126,98,110)(83,120,92,111)(84,121,93,112)(85,132,104,113)(86,133,105,114)(87,127,99,115)(88,128,100,116)(89,129,101,117)(90,130,102,118)(91,131,103,119)(134,162,150,178)(135,163,151,179)(136,164,152,180)(137,165,153,181)(138,166,154,182)(139,167,148,176)(140,168,149,177)(141,169,160,188)(142,170,161,189)(143,171,155,183)(144,172,156,184)(145,173,157,185)(146,174,158,186)(147,175,159,187), (1,134,39,150)(2,135,40,151)(3,136,41,152)(4,137,42,153)(5,138,36,154)(6,139,37,148)(7,140,38,149)(8,118,23,130)(9,119,24,131)(10,113,25,132)(11,114,26,133)(12,115,27,127)(13,116,28,128)(14,117,22,129)(15,125,221,109)(16,126,222,110)(17,120,223,111)(18,121,224,112)(19,122,218,106)(20,123,219,107)(21,124,220,108)(29,160,48,141)(30,161,49,142)(31,155,43,143)(32,156,44,144)(33,157,45,145)(34,158,46,146)(35,159,47,147)(50,178,66,162)(51,179,67,163)(52,180,68,164)(53,181,69,165)(54,182,70,166)(55,176,64,167)(56,177,65,168)(57,188,76,169)(58,189,77,170)(59,183,71,171)(60,184,72,172)(61,185,73,173)(62,186,74,174)(63,187,75,175)(78,206,94,190)(79,207,95,191)(80,208,96,192)(81,209,97,193)(82,210,98,194)(83,204,92,195)(84,205,93,196)(85,216,104,197)(86,217,105,198)(87,211,99,199)(88,212,100,200)(89,213,101,201)(90,214,102,202)(91,215,103,203)>;
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), (1,94,35,91)(2,95,29,85)(3,96,30,86)(4,97,31,87)(5,98,32,88)(6,92,33,89)(7,93,34,90)(8,177,18,174)(9,178,19,175)(10,179,20,169)(11,180,21,170)(12,181,15,171)(13,182,16,172)(14,176,17,173)(22,167,223,185)(23,168,224,186)(24,162,218,187)(25,163,219,188)(26,164,220,189)(27,165,221,183)(28,166,222,184)(36,82,44,100)(37,83,45,101)(38,84,46,102)(39,78,47,103)(40,79,48,104)(41,80,49,105)(42,81,43,99)(50,106,75,131)(51,107,76,132)(52,108,77,133)(53,109,71,127)(54,110,72,128)(55,111,73,129)(56,112,74,130)(57,113,67,123)(58,114,68,124)(59,115,69,125)(60,116,70,126)(61,117,64,120)(62,118,65,121)(63,119,66,122)(134,190,159,215)(135,191,160,216)(136,192,161,217)(137,193,155,211)(138,194,156,212)(139,195,157,213)(140,196,158,214)(141,197,151,207)(142,198,152,208)(143,199,153,209)(144,200,154,210)(145,201,148,204)(146,202,149,205)(147,203,150,206), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,224)(9,218)(10,219)(11,220)(12,221)(13,222)(14,223)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,101)(93,102)(94,103)(95,104)(96,105)(97,99)(98,100)(106,119)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118)(120,129)(121,130)(122,131)(123,132)(124,133)(125,127)(126,128)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,160)(142,161)(143,155)(144,156)(145,157)(146,158)(147,159)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,188)(170,189)(171,183)(172,184)(173,185)(174,186)(175,187)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,213)(205,214)(206,215)(207,216)(208,217)(209,211)(210,212), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,214,23,202)(9,215,24,203)(10,216,25,197)(11,217,26,198)(12,211,27,199)(13,212,28,200)(14,213,22,201)(15,193,221,209)(16,194,222,210)(17,195,223,204)(18,196,224,205)(19,190,218,206)(20,191,219,207)(21,192,220,208)(29,76,48,57)(30,77,49,58)(31,71,43,59)(32,72,44,60)(33,73,45,61)(34,74,46,62)(35,75,47,63)(78,122,94,106)(79,123,95,107)(80,124,96,108)(81,125,97,109)(82,126,98,110)(83,120,92,111)(84,121,93,112)(85,132,104,113)(86,133,105,114)(87,127,99,115)(88,128,100,116)(89,129,101,117)(90,130,102,118)(91,131,103,119)(134,162,150,178)(135,163,151,179)(136,164,152,180)(137,165,153,181)(138,166,154,182)(139,167,148,176)(140,168,149,177)(141,169,160,188)(142,170,161,189)(143,171,155,183)(144,172,156,184)(145,173,157,185)(146,174,158,186)(147,175,159,187), (1,134,39,150)(2,135,40,151)(3,136,41,152)(4,137,42,153)(5,138,36,154)(6,139,37,148)(7,140,38,149)(8,118,23,130)(9,119,24,131)(10,113,25,132)(11,114,26,133)(12,115,27,127)(13,116,28,128)(14,117,22,129)(15,125,221,109)(16,126,222,110)(17,120,223,111)(18,121,224,112)(19,122,218,106)(20,123,219,107)(21,124,220,108)(29,160,48,141)(30,161,49,142)(31,155,43,143)(32,156,44,144)(33,157,45,145)(34,158,46,146)(35,159,47,147)(50,178,66,162)(51,179,67,163)(52,180,68,164)(53,181,69,165)(54,182,70,166)(55,176,64,167)(56,177,65,168)(57,188,76,169)(58,189,77,170)(59,183,71,171)(60,184,72,172)(61,185,73,173)(62,186,74,174)(63,187,75,175)(78,206,94,190)(79,207,95,191)(80,208,96,192)(81,209,97,193)(82,210,98,194)(83,204,92,195)(84,205,93,196)(85,216,104,197)(86,217,105,198)(87,211,99,199)(88,212,100,200)(89,213,101,201)(90,214,102,202)(91,215,103,203) );
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)], [(1,94,35,91),(2,95,29,85),(3,96,30,86),(4,97,31,87),(5,98,32,88),(6,92,33,89),(7,93,34,90),(8,177,18,174),(9,178,19,175),(10,179,20,169),(11,180,21,170),(12,181,15,171),(13,182,16,172),(14,176,17,173),(22,167,223,185),(23,168,224,186),(24,162,218,187),(25,163,219,188),(26,164,220,189),(27,165,221,183),(28,166,222,184),(36,82,44,100),(37,83,45,101),(38,84,46,102),(39,78,47,103),(40,79,48,104),(41,80,49,105),(42,81,43,99),(50,106,75,131),(51,107,76,132),(52,108,77,133),(53,109,71,127),(54,110,72,128),(55,111,73,129),(56,112,74,130),(57,113,67,123),(58,114,68,124),(59,115,69,125),(60,116,70,126),(61,117,64,120),(62,118,65,121),(63,119,66,122),(134,190,159,215),(135,191,160,216),(136,192,161,217),(137,193,155,211),(138,194,156,212),(139,195,157,213),(140,196,158,214),(141,197,151,207),(142,198,152,208),(143,199,153,209),(144,200,154,210),(145,201,148,204),(146,202,149,205),(147,203,150,206)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,224),(9,218),(10,219),(11,220),(12,221),(13,222),(14,223),(15,27),(16,28),(17,22),(18,23),(19,24),(20,25),(21,26),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,76),(58,77),(59,71),(60,72),(61,73),(62,74),(63,75),(78,91),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(92,101),(93,102),(94,103),(95,104),(96,105),(97,99),(98,100),(106,119),(107,113),(108,114),(109,115),(110,116),(111,117),(112,118),(120,129),(121,130),(122,131),(123,132),(124,133),(125,127),(126,128),(134,150),(135,151),(136,152),(137,153),(138,154),(139,148),(140,149),(141,160),(142,161),(143,155),(144,156),(145,157),(146,158),(147,159),(162,178),(163,179),(164,180),(165,181),(166,182),(167,176),(168,177),(169,188),(170,189),(171,183),(172,184),(173,185),(174,186),(175,187),(190,203),(191,197),(192,198),(193,199),(194,200),(195,201),(196,202),(204,213),(205,214),(206,215),(207,216),(208,217),(209,211),(210,212)], [(1,50,39,66),(2,51,40,67),(3,52,41,68),(4,53,42,69),(5,54,36,70),(6,55,37,64),(7,56,38,65),(8,214,23,202),(9,215,24,203),(10,216,25,197),(11,217,26,198),(12,211,27,199),(13,212,28,200),(14,213,22,201),(15,193,221,209),(16,194,222,210),(17,195,223,204),(18,196,224,205),(19,190,218,206),(20,191,219,207),(21,192,220,208),(29,76,48,57),(30,77,49,58),(31,71,43,59),(32,72,44,60),(33,73,45,61),(34,74,46,62),(35,75,47,63),(78,122,94,106),(79,123,95,107),(80,124,96,108),(81,125,97,109),(82,126,98,110),(83,120,92,111),(84,121,93,112),(85,132,104,113),(86,133,105,114),(87,127,99,115),(88,128,100,116),(89,129,101,117),(90,130,102,118),(91,131,103,119),(134,162,150,178),(135,163,151,179),(136,164,152,180),(137,165,153,181),(138,166,154,182),(139,167,148,176),(140,168,149,177),(141,169,160,188),(142,170,161,189),(143,171,155,183),(144,172,156,184),(145,173,157,185),(146,174,158,186),(147,175,159,187)], [(1,134,39,150),(2,135,40,151),(3,136,41,152),(4,137,42,153),(5,138,36,154),(6,139,37,148),(7,140,38,149),(8,118,23,130),(9,119,24,131),(10,113,25,132),(11,114,26,133),(12,115,27,127),(13,116,28,128),(14,117,22,129),(15,125,221,109),(16,126,222,110),(17,120,223,111),(18,121,224,112),(19,122,218,106),(20,123,219,107),(21,124,220,108),(29,160,48,141),(30,161,49,142),(31,155,43,143),(32,156,44,144),(33,157,45,145),(34,158,46,146),(35,159,47,147),(50,178,66,162),(51,179,67,163),(52,180,68,164),(53,181,69,165),(54,182,70,166),(55,176,64,167),(56,177,65,168),(57,188,76,169),(58,189,77,170),(59,183,71,171),(60,184,72,172),(61,185,73,173),(62,186,74,174),(63,187,75,175),(78,206,94,190),(79,207,95,191),(80,208,96,192),(81,209,97,193),(82,210,98,194),(83,204,92,195),(84,205,93,196),(85,216,104,197),(86,217,105,198),(87,211,99,199),(88,212,100,200),(89,213,101,201),(90,214,102,202),(91,215,103,203)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 28 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 28 |
| 28 | 27 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [25,0,0,0,0,25,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,0,0,0,0,1,28,0,0,2,28],[28,0,0,0,0,28,0,0,0,0,1,28,0,0,0,28],[28,1,0,0,27,1,0,0,0,0,1,0,0,0,0,1],[17,12,0,0,0,12,0,0,0,0,28,0,0,0,0,28] >;
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4Q | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AP | 28A | ··· | 28AV | 28AW | ··· | 28CX |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | Q8 | D4 | C7×Q8 | C7×D4 | 2- (1+4) | C7×2- (1+4) |
| kernel | C7×D4×Q8 | D4×C28 | Q8×C28 | C7×C22⋊Q8 | C7×C4⋊Q8 | Q8×C2×C14 | D4×Q8 | C4×D4 | C4×Q8 | C22⋊Q8 | C4⋊Q8 | C22×Q8 | C7×D4 | C7×Q8 | D4 | Q8 | C14 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 6 | 18 | 6 | 36 | 18 | 12 | 4 | 4 | 24 | 24 | 1 | 6 |
In GAP, Magma, Sage, TeX
C_7\times D_4\times Q_8
% in TeX
G:=Group("C7xD4xQ8"); // GroupNames label
G:=SmallGroup(448,1332);
// by ID
G=gap.SmallGroup(448,1332);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,792,4790,1690,416]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,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=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀