direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×Q8⋊5D4, C14.1182- 1+4, Q8⋊5(C7×D4), (C7×Q8)⋊23D4, (D4×C28)⋊46C2, (C4×D4)⋊17C14, (C4×Q8)⋊11C14, (Q8×C28)⋊31C2, C4.42(D4×C14), C4⋊D4⋊13C14, C28.403(C2×D4), C22⋊Q8⋊13C14, (C22×Q8)⋊6C14, C4.4D4⋊11C14, C42.43(C2×C14), (C2×C28).714C23, (C2×C14).368C24, (C4×C28).284C22, C14.196(C22×D4), (D4×C14).322C22, C23.42(C22×C14), C22.42(C23×C14), (Q8×C14).274C22, C2.10(C7×2- 1+4), (C22×C14).100C23, (C22×C28).454C22, (Q8×C2×C14)⋊18C2, C2.20(D4×C2×C14), (C2×C4○D4)⋊8C14, C22⋊3(C7×C4○D4), (C14×C4○D4)⋊24C2, (C7×C4⋊D4)⋊40C2, C4⋊C4.71(C2×C14), C2.22(C14×C4○D4), (C7×C22⋊Q8)⋊40C2, (C2×C14)⋊17(C4○D4), (C2×D4).67(C2×C14), C14.241(C2×C4○D4), (C7×C4.4D4)⋊31C2, (C2×Q8).61(C2×C14), C22⋊C4.19(C2×C14), (C7×C4⋊C4).396C22, (C22×C4).66(C2×C14), (C2×C4).60(C22×C14), (C7×C22⋊C4).88C22, SmallGroup(448,1331)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Q8⋊5D4
G = < a,b,c,d,e | a7=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=ece=b2c, ede=d-1 >
Subgroups: 426 in 290 conjugacy classes, 166 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C2×C14, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, Q8⋊5D4, C4×C28, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, Q8×C14, Q8×C14, C7×C4○D4, D4×C28, Q8×C28, C7×C4⋊D4, C7×C22⋊Q8, C7×C4.4D4, Q8×C2×C14, C14×C4○D4, C7×Q8⋊5D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C24, C2×C14, C22×D4, C2×C4○D4, 2- 1+4, C7×D4, C22×C14, Q8⋊5D4, D4×C14, C7×C4○D4, C23×C14, D4×C2×C14, C14×C4○D4, C7×2- 1+4, C7×Q8⋊5D4
►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 70 34 63)(2 64 35 57)(3 65 29 58)(4 66 30 59)(5 67 31 60)(6 68 32 61)(7 69 33 62)(8 204 219 197)(9 205 220 198)(10 206 221 199)(11 207 222 200)(12 208 223 201)(13 209 224 202)(14 210 218 203)(15 195 22 212)(16 196 23 213)(17 190 24 214)(18 191 25 215)(19 192 26 216)(20 193 27 217)(21 194 28 211)(36 75 46 51)(37 76 47 52)(38 77 48 53)(39 71 49 54)(40 72 43 55)(41 73 44 56)(42 74 45 50)(78 126 102 119)(79 120 103 113)(80 121 104 114)(81 122 105 115)(82 123 99 116)(83 124 100 117)(84 125 101 118)(85 131 92 107)(86 132 93 108)(87 133 94 109)(88 127 95 110)(89 128 96 111)(90 129 97 112)(91 130 98 106)(134 175 158 182)(135 169 159 176)(136 170 160 177)(137 171 161 178)(138 172 155 179)(139 173 156 180)(140 174 157 181)(141 163 148 187)(142 164 149 188)(143 165 150 189)(144 166 151 183)(145 167 152 184)(146 168 153 185)(147 162 154 186)
(1 98 34 91)(2 92 35 85)(3 93 29 86)(4 94 30 87)(5 95 31 88)(6 96 32 89)(7 97 33 90)(8 176 219 169)(9 177 220 170)(10 178 221 171)(11 179 222 172)(12 180 223 173)(13 181 224 174)(14 182 218 175)(15 167 22 184)(16 168 23 185)(17 162 24 186)(18 163 25 187)(19 164 26 188)(20 165 27 189)(21 166 28 183)(36 103 46 79)(37 104 47 80)(38 105 48 81)(39 99 49 82)(40 100 43 83)(41 101 44 84)(42 102 45 78)(50 119 74 126)(51 113 75 120)(52 114 76 121)(53 115 77 122)(54 116 71 123)(55 117 72 124)(56 118 73 125)(57 107 64 131)(58 108 65 132)(59 109 66 133)(60 110 67 127)(61 111 68 128)(62 112 69 129)(63 106 70 130)(134 210 158 203)(135 204 159 197)(136 205 160 198)(137 206 161 199)(138 207 155 200)(139 208 156 201)(140 209 157 202)(141 215 148 191)(142 216 149 192)(143 217 150 193)(144 211 151 194)(145 212 152 195)(146 213 153 196)(147 214 154 190)
(1 147 45 158)(2 141 46 159)(3 142 47 160)(4 143 48 161)(5 144 49 155)(6 145 43 156)(7 146 44 157)(8 131 18 113)(9 132 19 114)(10 133 20 115)(11 127 21 116)(12 128 15 117)(13 129 16 118)(14 130 17 119)(22 124 223 111)(23 125 224 112)(24 126 218 106)(25 120 219 107)(26 121 220 108)(27 122 221 109)(28 123 222 110)(29 149 37 136)(30 150 38 137)(31 151 39 138)(32 152 40 139)(33 153 41 140)(34 154 42 134)(35 148 36 135)(50 182 70 162)(51 176 64 163)(52 177 65 164)(53 178 66 165)(54 179 67 166)(55 180 68 167)(56 181 69 168)(57 187 75 169)(58 188 76 170)(59 189 77 171)(60 183 71 172)(61 184 72 173)(62 185 73 174)(63 186 74 175)(78 210 98 190)(79 204 92 191)(80 205 93 192)(81 206 94 193)(82 207 95 194)(83 208 96 195)(84 209 97 196)(85 215 103 197)(86 216 104 198)(87 217 105 199)(88 211 99 200)(89 212 100 201)(90 213 101 202)(91 214 102 203)
(1 158)(2 159)(3 160)(4 161)(5 155)(6 156)(7 157)(8 131)(9 132)(10 133)(11 127)(12 128)(13 129)(14 130)(15 117)(16 118)(17 119)(18 113)(19 114)(20 115)(21 116)(22 124)(23 125)(24 126)(25 120)(26 121)(27 122)(28 123)(29 136)(30 137)(31 138)(32 139)(33 140)(34 134)(35 135)(36 148)(37 149)(38 150)(39 151)(40 152)(41 153)(42 154)(43 145)(44 146)(45 147)(46 141)(47 142)(48 143)(49 144)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
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), (1,70,34,63)(2,64,35,57)(3,65,29,58)(4,66,30,59)(5,67,31,60)(6,68,32,61)(7,69,33,62)(8,204,219,197)(9,205,220,198)(10,206,221,199)(11,207,222,200)(12,208,223,201)(13,209,224,202)(14,210,218,203)(15,195,22,212)(16,196,23,213)(17,190,24,214)(18,191,25,215)(19,192,26,216)(20,193,27,217)(21,194,28,211)(36,75,46,51)(37,76,47,52)(38,77,48,53)(39,71,49,54)(40,72,43,55)(41,73,44,56)(42,74,45,50)(78,126,102,119)(79,120,103,113)(80,121,104,114)(81,122,105,115)(82,123,99,116)(83,124,100,117)(84,125,101,118)(85,131,92,107)(86,132,93,108)(87,133,94,109)(88,127,95,110)(89,128,96,111)(90,129,97,112)(91,130,98,106)(134,175,158,182)(135,169,159,176)(136,170,160,177)(137,171,161,178)(138,172,155,179)(139,173,156,180)(140,174,157,181)(141,163,148,187)(142,164,149,188)(143,165,150,189)(144,166,151,183)(145,167,152,184)(146,168,153,185)(147,162,154,186), (1,98,34,91)(2,92,35,85)(3,93,29,86)(4,94,30,87)(5,95,31,88)(6,96,32,89)(7,97,33,90)(8,176,219,169)(9,177,220,170)(10,178,221,171)(11,179,222,172)(12,180,223,173)(13,181,224,174)(14,182,218,175)(15,167,22,184)(16,168,23,185)(17,162,24,186)(18,163,25,187)(19,164,26,188)(20,165,27,189)(21,166,28,183)(36,103,46,79)(37,104,47,80)(38,105,48,81)(39,99,49,82)(40,100,43,83)(41,101,44,84)(42,102,45,78)(50,119,74,126)(51,113,75,120)(52,114,76,121)(53,115,77,122)(54,116,71,123)(55,117,72,124)(56,118,73,125)(57,107,64,131)(58,108,65,132)(59,109,66,133)(60,110,67,127)(61,111,68,128)(62,112,69,129)(63,106,70,130)(134,210,158,203)(135,204,159,197)(136,205,160,198)(137,206,161,199)(138,207,155,200)(139,208,156,201)(140,209,157,202)(141,215,148,191)(142,216,149,192)(143,217,150,193)(144,211,151,194)(145,212,152,195)(146,213,153,196)(147,214,154,190), (1,147,45,158)(2,141,46,159)(3,142,47,160)(4,143,48,161)(5,144,49,155)(6,145,43,156)(7,146,44,157)(8,131,18,113)(9,132,19,114)(10,133,20,115)(11,127,21,116)(12,128,15,117)(13,129,16,118)(14,130,17,119)(22,124,223,111)(23,125,224,112)(24,126,218,106)(25,120,219,107)(26,121,220,108)(27,122,221,109)(28,123,222,110)(29,149,37,136)(30,150,38,137)(31,151,39,138)(32,152,40,139)(33,153,41,140)(34,154,42,134)(35,148,36,135)(50,182,70,162)(51,176,64,163)(52,177,65,164)(53,178,66,165)(54,179,67,166)(55,180,68,167)(56,181,69,168)(57,187,75,169)(58,188,76,170)(59,189,77,171)(60,183,71,172)(61,184,72,173)(62,185,73,174)(63,186,74,175)(78,210,98,190)(79,204,92,191)(80,205,93,192)(81,206,94,193)(82,207,95,194)(83,208,96,195)(84,209,97,196)(85,215,103,197)(86,216,104,198)(87,217,105,199)(88,211,99,200)(89,212,100,201)(90,213,101,202)(91,214,102,203), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,131)(9,132)(10,133)(11,127)(12,128)(13,129)(14,130)(15,117)(16,118)(17,119)(18,113)(19,114)(20,115)(21,116)(22,124)(23,125)(24,126)(25,120)(26,121)(27,122)(28,123)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,145)(44,146)(45,147)(46,141)(47,142)(48,143)(49,144)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224)>;
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), (1,70,34,63)(2,64,35,57)(3,65,29,58)(4,66,30,59)(5,67,31,60)(6,68,32,61)(7,69,33,62)(8,204,219,197)(9,205,220,198)(10,206,221,199)(11,207,222,200)(12,208,223,201)(13,209,224,202)(14,210,218,203)(15,195,22,212)(16,196,23,213)(17,190,24,214)(18,191,25,215)(19,192,26,216)(20,193,27,217)(21,194,28,211)(36,75,46,51)(37,76,47,52)(38,77,48,53)(39,71,49,54)(40,72,43,55)(41,73,44,56)(42,74,45,50)(78,126,102,119)(79,120,103,113)(80,121,104,114)(81,122,105,115)(82,123,99,116)(83,124,100,117)(84,125,101,118)(85,131,92,107)(86,132,93,108)(87,133,94,109)(88,127,95,110)(89,128,96,111)(90,129,97,112)(91,130,98,106)(134,175,158,182)(135,169,159,176)(136,170,160,177)(137,171,161,178)(138,172,155,179)(139,173,156,180)(140,174,157,181)(141,163,148,187)(142,164,149,188)(143,165,150,189)(144,166,151,183)(145,167,152,184)(146,168,153,185)(147,162,154,186), (1,98,34,91)(2,92,35,85)(3,93,29,86)(4,94,30,87)(5,95,31,88)(6,96,32,89)(7,97,33,90)(8,176,219,169)(9,177,220,170)(10,178,221,171)(11,179,222,172)(12,180,223,173)(13,181,224,174)(14,182,218,175)(15,167,22,184)(16,168,23,185)(17,162,24,186)(18,163,25,187)(19,164,26,188)(20,165,27,189)(21,166,28,183)(36,103,46,79)(37,104,47,80)(38,105,48,81)(39,99,49,82)(40,100,43,83)(41,101,44,84)(42,102,45,78)(50,119,74,126)(51,113,75,120)(52,114,76,121)(53,115,77,122)(54,116,71,123)(55,117,72,124)(56,118,73,125)(57,107,64,131)(58,108,65,132)(59,109,66,133)(60,110,67,127)(61,111,68,128)(62,112,69,129)(63,106,70,130)(134,210,158,203)(135,204,159,197)(136,205,160,198)(137,206,161,199)(138,207,155,200)(139,208,156,201)(140,209,157,202)(141,215,148,191)(142,216,149,192)(143,217,150,193)(144,211,151,194)(145,212,152,195)(146,213,153,196)(147,214,154,190), (1,147,45,158)(2,141,46,159)(3,142,47,160)(4,143,48,161)(5,144,49,155)(6,145,43,156)(7,146,44,157)(8,131,18,113)(9,132,19,114)(10,133,20,115)(11,127,21,116)(12,128,15,117)(13,129,16,118)(14,130,17,119)(22,124,223,111)(23,125,224,112)(24,126,218,106)(25,120,219,107)(26,121,220,108)(27,122,221,109)(28,123,222,110)(29,149,37,136)(30,150,38,137)(31,151,39,138)(32,152,40,139)(33,153,41,140)(34,154,42,134)(35,148,36,135)(50,182,70,162)(51,176,64,163)(52,177,65,164)(53,178,66,165)(54,179,67,166)(55,180,68,167)(56,181,69,168)(57,187,75,169)(58,188,76,170)(59,189,77,171)(60,183,71,172)(61,184,72,173)(62,185,73,174)(63,186,74,175)(78,210,98,190)(79,204,92,191)(80,205,93,192)(81,206,94,193)(82,207,95,194)(83,208,96,195)(84,209,97,196)(85,215,103,197)(86,216,104,198)(87,217,105,199)(88,211,99,200)(89,212,100,201)(90,213,101,202)(91,214,102,203), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,131)(9,132)(10,133)(11,127)(12,128)(13,129)(14,130)(15,117)(16,118)(17,119)(18,113)(19,114)(20,115)(21,116)(22,124)(23,125)(24,126)(25,120)(26,121)(27,122)(28,123)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,145)(44,146)(45,147)(46,141)(47,142)(48,143)(49,144)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224) );
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)], [(1,70,34,63),(2,64,35,57),(3,65,29,58),(4,66,30,59),(5,67,31,60),(6,68,32,61),(7,69,33,62),(8,204,219,197),(9,205,220,198),(10,206,221,199),(11,207,222,200),(12,208,223,201),(13,209,224,202),(14,210,218,203),(15,195,22,212),(16,196,23,213),(17,190,24,214),(18,191,25,215),(19,192,26,216),(20,193,27,217),(21,194,28,211),(36,75,46,51),(37,76,47,52),(38,77,48,53),(39,71,49,54),(40,72,43,55),(41,73,44,56),(42,74,45,50),(78,126,102,119),(79,120,103,113),(80,121,104,114),(81,122,105,115),(82,123,99,116),(83,124,100,117),(84,125,101,118),(85,131,92,107),(86,132,93,108),(87,133,94,109),(88,127,95,110),(89,128,96,111),(90,129,97,112),(91,130,98,106),(134,175,158,182),(135,169,159,176),(136,170,160,177),(137,171,161,178),(138,172,155,179),(139,173,156,180),(140,174,157,181),(141,163,148,187),(142,164,149,188),(143,165,150,189),(144,166,151,183),(145,167,152,184),(146,168,153,185),(147,162,154,186)], [(1,98,34,91),(2,92,35,85),(3,93,29,86),(4,94,30,87),(5,95,31,88),(6,96,32,89),(7,97,33,90),(8,176,219,169),(9,177,220,170),(10,178,221,171),(11,179,222,172),(12,180,223,173),(13,181,224,174),(14,182,218,175),(15,167,22,184),(16,168,23,185),(17,162,24,186),(18,163,25,187),(19,164,26,188),(20,165,27,189),(21,166,28,183),(36,103,46,79),(37,104,47,80),(38,105,48,81),(39,99,49,82),(40,100,43,83),(41,101,44,84),(42,102,45,78),(50,119,74,126),(51,113,75,120),(52,114,76,121),(53,115,77,122),(54,116,71,123),(55,117,72,124),(56,118,73,125),(57,107,64,131),(58,108,65,132),(59,109,66,133),(60,110,67,127),(61,111,68,128),(62,112,69,129),(63,106,70,130),(134,210,158,203),(135,204,159,197),(136,205,160,198),(137,206,161,199),(138,207,155,200),(139,208,156,201),(140,209,157,202),(141,215,148,191),(142,216,149,192),(143,217,150,193),(144,211,151,194),(145,212,152,195),(146,213,153,196),(147,214,154,190)], [(1,147,45,158),(2,141,46,159),(3,142,47,160),(4,143,48,161),(5,144,49,155),(6,145,43,156),(7,146,44,157),(8,131,18,113),(9,132,19,114),(10,133,20,115),(11,127,21,116),(12,128,15,117),(13,129,16,118),(14,130,17,119),(22,124,223,111),(23,125,224,112),(24,126,218,106),(25,120,219,107),(26,121,220,108),(27,122,221,109),(28,123,222,110),(29,149,37,136),(30,150,38,137),(31,151,39,138),(32,152,40,139),(33,153,41,140),(34,154,42,134),(35,148,36,135),(50,182,70,162),(51,176,64,163),(52,177,65,164),(53,178,66,165),(54,179,67,166),(55,180,68,167),(56,181,69,168),(57,187,75,169),(58,188,76,170),(59,189,77,171),(60,183,71,172),(61,184,72,173),(62,185,73,174),(63,186,74,175),(78,210,98,190),(79,204,92,191),(80,205,93,192),(81,206,94,193),(82,207,95,194),(83,208,96,195),(84,209,97,196),(85,215,103,197),(86,216,104,198),(87,217,105,199),(88,211,99,200),(89,212,100,201),(90,213,101,202),(91,214,102,203)], [(1,158),(2,159),(3,160),(4,161),(5,155),(6,156),(7,157),(8,131),(9,132),(10,133),(11,127),(12,128),(13,129),(14,130),(15,117),(16,118),(17,119),(18,113),(19,114),(20,115),(21,116),(22,124),(23,125),(24,126),(25,120),(26,121),(27,122),(28,123),(29,136),(30,137),(31,138),(32,139),(33,140),(34,134),(35,135),(36,148),(37,149),(38,150),(39,151),(40,152),(41,153),(42,154),(43,145),(44,146),(45,147),(46,141),(47,142),(48,143),(49,144),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)]])
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4J | 4K | ··· | 4P | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AV | 28A | ··· | 28BH | 28BI | ··· | 28CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | C4○D4 | C7×D4 | C7×C4○D4 | 2- 1+4 | C7×2- 1+4 |
| kernel | C7×Q8⋊5D4 | D4×C28 | Q8×C28 | C7×C4⋊D4 | C7×C22⋊Q8 | C7×C4.4D4 | Q8×C2×C14 | C14×C4○D4 | Q8⋊5D4 | C4×D4 | C4×Q8 | C4⋊D4 | C22⋊Q8 | C4.4D4 | C22×Q8 | C2×C4○D4 | C7×Q8 | C2×C14 | Q8 | C22 | C14 | C2 |
| # reps | 1 | 3 | 1 | 3 | 3 | 3 | 1 | 1 | 6 | 18 | 6 | 18 | 18 | 18 | 6 | 6 | 4 | 4 | 24 | 24 | 1 | 6 |
Matrix representation of C7×Q8⋊5D4 ►in GL4(𝔽29) generated by
| 23 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 27 |
| 0 | 0 | 13 | 24 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 4 | 1 |
| 0 | 0 | 12 | 25 |
| 18 | 27 | 0 | 0 |
| 3 | 11 | 0 | 0 |
| 0 | 0 | 27 | 24 |
| 0 | 0 | 18 | 2 |
| 18 | 27 | 0 | 0 |
| 2 | 11 | 0 | 0 |
| 0 | 0 | 27 | 24 |
| 0 | 0 | 18 | 2 |
G:=sub<GL(4,GF(29))| [23,0,0,0,0,23,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,5,13,0,0,27,24],[28,0,0,0,0,28,0,0,0,0,4,12,0,0,1,25],[18,3,0,0,27,11,0,0,0,0,27,18,0,0,24,2],[18,2,0,0,27,11,0,0,0,0,27,18,0,0,24,2] >;
C7×Q8⋊5D4 in GAP, Magma, Sage, TeX
C_7\times Q_8\rtimes_5D_4
% in TeX
G:=Group("C7xQ8:5D4"); // GroupNames label
G:=SmallGroup(448,1331);
// by ID
G=gap.SmallGroup(448,1331);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,792,4790,1690,416]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations