direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×Q8⋊5D4, C14.1182- (1+4), Q8⋊5(C7×D4), (C7×Q8)⋊23D4, (C4×D4)⋊17C14, (D4×C28)⋊46C2, (Q8×C28)⋊31C2, (C4×Q8)⋊11C14, 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×C28).454C22, (C22×C14).100C23, (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), (C7×C4⋊C4).396C22, C22⋊C4.19(C2×C14), (C2×C4).60(C22×C14), (C22×C4).66(C2×C14), (C7×C22⋊C4).88C22, SmallGroup(448,1331)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 426 in 290 conjugacy classes, 166 normal (28 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×8], C22, C22 [×2], C22 [×11], C7, C2×C4 [×2], C2×C4 [×9], C2×C4 [×12], D4 [×12], Q8 [×4], Q8 [×6], C23, C23 [×3], C14 [×3], C14 [×5], C42 [×3], C22⋊C4, C22⋊C4 [×9], C4⋊C4 [×6], C22×C4 [×6], C2×D4 [×6], C2×Q8, C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×4], C28 [×6], C28 [×8], C2×C14, C2×C14 [×2], C2×C14 [×11], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, C2×C28 [×2], C2×C28 [×9], C2×C28 [×12], C7×D4 [×12], C7×Q8 [×4], C7×Q8 [×6], C22×C14, C22×C14 [×3], Q8⋊5D4, C4×C28 [×3], C7×C22⋊C4, C7×C22⋊C4 [×9], C7×C4⋊C4 [×6], C22×C28 [×6], D4×C14 [×6], Q8×C14, Q8×C14 [×3], Q8×C14 [×4], C7×C4○D4 [×4], D4×C28 [×3], Q8×C28, C7×C4⋊D4 [×3], C7×C22⋊Q8 [×3], C7×C4.4D4 [×3], Q8×C2×C14, C14×C4○D4, C7×Q8⋊5D4
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C4○D4 [×2], C24, C2×C14 [×35], C22×D4, C2×C4○D4, 2- (1+4), C7×D4 [×4], C22×C14 [×15], Q8⋊5D4, D4×C14 [×6], C7×C4○D4 [×2], C23×C14, D4×C2×C14, C14×C4○D4, C7×2- (1+4), C7×Q8⋊5D4
Generators and relations
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 >
►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 75 39 63)(2 76 40 57)(3 77 41 58)(4 71 42 59)(5 72 36 60)(6 73 37 61)(7 74 38 62)(8 196 23 205)(9 190 24 206)(10 191 25 207)(11 192 26 208)(12 193 27 209)(13 194 28 210)(14 195 22 204)(15 211 221 199)(16 212 222 200)(17 213 223 201)(18 214 224 202)(19 215 218 203)(20 216 219 197)(21 217 220 198)(29 51 48 67)(30 52 49 68)(31 53 43 69)(32 54 44 70)(33 55 45 64)(34 56 46 65)(35 50 47 66)(78 131 94 119)(79 132 95 113)(80 133 96 114)(81 127 97 115)(82 128 98 116)(83 129 92 117)(84 130 93 118)(85 123 104 107)(86 124 105 108)(87 125 99 109)(88 126 100 110)(89 120 101 111)(90 121 102 112)(91 122 103 106)(134 175 150 187)(135 169 151 188)(136 170 152 189)(137 171 153 183)(138 172 154 184)(139 173 148 185)(140 174 149 186)(141 163 160 179)(142 164 161 180)(143 165 155 181)(144 166 156 182)(145 167 157 176)(146 168 158 177)(147 162 159 178)
(1 103 39 91)(2 104 40 85)(3 105 41 86)(4 99 42 87)(5 100 36 88)(6 101 37 89)(7 102 38 90)(8 168 23 177)(9 162 24 178)(10 163 25 179)(11 164 26 180)(12 165 27 181)(13 166 28 182)(14 167 22 176)(15 183 221 171)(16 184 222 172)(17 185 223 173)(18 186 224 174)(19 187 218 175)(20 188 219 169)(21 189 220 170)(29 79 48 95)(30 80 49 96)(31 81 43 97)(32 82 44 98)(33 83 45 92)(34 84 46 93)(35 78 47 94)(50 119 66 131)(51 113 67 132)(52 114 68 133)(53 115 69 127)(54 116 70 128)(55 117 64 129)(56 118 65 130)(57 107 76 123)(58 108 77 124)(59 109 71 125)(60 110 72 126)(61 111 73 120)(62 112 74 121)(63 106 75 122)(134 215 150 203)(135 216 151 197)(136 217 152 198)(137 211 153 199)(138 212 154 200)(139 213 148 201)(140 214 149 202)(141 207 160 191)(142 208 161 192)(143 209 155 193)(144 210 156 194)(145 204 157 195)(146 205 158 196)(147 206 159 190)
(1 147 35 150)(2 141 29 151)(3 142 30 152)(4 143 31 153)(5 144 32 154)(6 145 33 148)(7 146 34 149)(8 118 18 121)(9 119 19 122)(10 113 20 123)(11 114 21 124)(12 115 15 125)(13 116 16 126)(14 117 17 120)(22 129 223 111)(23 130 224 112)(24 131 218 106)(25 132 219 107)(26 133 220 108)(27 127 221 109)(28 128 222 110)(36 156 44 138)(37 157 45 139)(38 158 46 140)(39 159 47 134)(40 160 48 135)(41 161 49 136)(42 155 43 137)(50 187 75 162)(51 188 76 163)(52 189 77 164)(53 183 71 165)(54 184 72 166)(55 185 73 167)(56 186 74 168)(57 179 67 169)(58 180 68 170)(59 181 69 171)(60 182 70 172)(61 176 64 173)(62 177 65 174)(63 178 66 175)(78 215 103 190)(79 216 104 191)(80 217 105 192)(81 211 99 193)(82 212 100 194)(83 213 101 195)(84 214 102 196)(85 207 95 197)(86 208 96 198)(87 209 97 199)(88 210 98 200)(89 204 92 201)(90 205 93 202)(91 206 94 203)
(1 150)(2 151)(3 152)(4 153)(5 154)(6 148)(7 149)(8 118)(9 119)(10 113)(11 114)(12 115)(13 116)(14 117)(15 125)(16 126)(17 120)(18 121)(19 122)(20 123)(21 124)(22 129)(23 130)(24 131)(25 132)(26 133)(27 127)(28 128)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 138)(37 139)(38 140)(39 134)(40 135)(41 136)(42 137)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 161)(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), (1,75,39,63)(2,76,40,57)(3,77,41,58)(4,71,42,59)(5,72,36,60)(6,73,37,61)(7,74,38,62)(8,196,23,205)(9,190,24,206)(10,191,25,207)(11,192,26,208)(12,193,27,209)(13,194,28,210)(14,195,22,204)(15,211,221,199)(16,212,222,200)(17,213,223,201)(18,214,224,202)(19,215,218,203)(20,216,219,197)(21,217,220,198)(29,51,48,67)(30,52,49,68)(31,53,43,69)(32,54,44,70)(33,55,45,64)(34,56,46,65)(35,50,47,66)(78,131,94,119)(79,132,95,113)(80,133,96,114)(81,127,97,115)(82,128,98,116)(83,129,92,117)(84,130,93,118)(85,123,104,107)(86,124,105,108)(87,125,99,109)(88,126,100,110)(89,120,101,111)(90,121,102,112)(91,122,103,106)(134,175,150,187)(135,169,151,188)(136,170,152,189)(137,171,153,183)(138,172,154,184)(139,173,148,185)(140,174,149,186)(141,163,160,179)(142,164,161,180)(143,165,155,181)(144,166,156,182)(145,167,157,176)(146,168,158,177)(147,162,159,178), (1,103,39,91)(2,104,40,85)(3,105,41,86)(4,99,42,87)(5,100,36,88)(6,101,37,89)(7,102,38,90)(8,168,23,177)(9,162,24,178)(10,163,25,179)(11,164,26,180)(12,165,27,181)(13,166,28,182)(14,167,22,176)(15,183,221,171)(16,184,222,172)(17,185,223,173)(18,186,224,174)(19,187,218,175)(20,188,219,169)(21,189,220,170)(29,79,48,95)(30,80,49,96)(31,81,43,97)(32,82,44,98)(33,83,45,92)(34,84,46,93)(35,78,47,94)(50,119,66,131)(51,113,67,132)(52,114,68,133)(53,115,69,127)(54,116,70,128)(55,117,64,129)(56,118,65,130)(57,107,76,123)(58,108,77,124)(59,109,71,125)(60,110,72,126)(61,111,73,120)(62,112,74,121)(63,106,75,122)(134,215,150,203)(135,216,151,197)(136,217,152,198)(137,211,153,199)(138,212,154,200)(139,213,148,201)(140,214,149,202)(141,207,160,191)(142,208,161,192)(143,209,155,193)(144,210,156,194)(145,204,157,195)(146,205,158,196)(147,206,159,190), (1,147,35,150)(2,141,29,151)(3,142,30,152)(4,143,31,153)(5,144,32,154)(6,145,33,148)(7,146,34,149)(8,118,18,121)(9,119,19,122)(10,113,20,123)(11,114,21,124)(12,115,15,125)(13,116,16,126)(14,117,17,120)(22,129,223,111)(23,130,224,112)(24,131,218,106)(25,132,219,107)(26,133,220,108)(27,127,221,109)(28,128,222,110)(36,156,44,138)(37,157,45,139)(38,158,46,140)(39,159,47,134)(40,160,48,135)(41,161,49,136)(42,155,43,137)(50,187,75,162)(51,188,76,163)(52,189,77,164)(53,183,71,165)(54,184,72,166)(55,185,73,167)(56,186,74,168)(57,179,67,169)(58,180,68,170)(59,181,69,171)(60,182,70,172)(61,176,64,173)(62,177,65,174)(63,178,66,175)(78,215,103,190)(79,216,104,191)(80,217,105,192)(81,211,99,193)(82,212,100,194)(83,213,101,195)(84,214,102,196)(85,207,95,197)(86,208,96,198)(87,209,97,199)(88,210,98,200)(89,204,92,201)(90,205,93,202)(91,206,94,203), (1,150)(2,151)(3,152)(4,153)(5,154)(6,148)(7,149)(8,118)(9,119)(10,113)(11,114)(12,115)(13,116)(14,117)(15,125)(16,126)(17,120)(18,121)(19,122)(20,123)(21,124)(22,129)(23,130)(24,131)(25,132)(26,133)(27,127)(28,128)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,138)(37,139)(38,140)(39,134)(40,135)(41,136)(42,137)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(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), (1,75,39,63)(2,76,40,57)(3,77,41,58)(4,71,42,59)(5,72,36,60)(6,73,37,61)(7,74,38,62)(8,196,23,205)(9,190,24,206)(10,191,25,207)(11,192,26,208)(12,193,27,209)(13,194,28,210)(14,195,22,204)(15,211,221,199)(16,212,222,200)(17,213,223,201)(18,214,224,202)(19,215,218,203)(20,216,219,197)(21,217,220,198)(29,51,48,67)(30,52,49,68)(31,53,43,69)(32,54,44,70)(33,55,45,64)(34,56,46,65)(35,50,47,66)(78,131,94,119)(79,132,95,113)(80,133,96,114)(81,127,97,115)(82,128,98,116)(83,129,92,117)(84,130,93,118)(85,123,104,107)(86,124,105,108)(87,125,99,109)(88,126,100,110)(89,120,101,111)(90,121,102,112)(91,122,103,106)(134,175,150,187)(135,169,151,188)(136,170,152,189)(137,171,153,183)(138,172,154,184)(139,173,148,185)(140,174,149,186)(141,163,160,179)(142,164,161,180)(143,165,155,181)(144,166,156,182)(145,167,157,176)(146,168,158,177)(147,162,159,178), (1,103,39,91)(2,104,40,85)(3,105,41,86)(4,99,42,87)(5,100,36,88)(6,101,37,89)(7,102,38,90)(8,168,23,177)(9,162,24,178)(10,163,25,179)(11,164,26,180)(12,165,27,181)(13,166,28,182)(14,167,22,176)(15,183,221,171)(16,184,222,172)(17,185,223,173)(18,186,224,174)(19,187,218,175)(20,188,219,169)(21,189,220,170)(29,79,48,95)(30,80,49,96)(31,81,43,97)(32,82,44,98)(33,83,45,92)(34,84,46,93)(35,78,47,94)(50,119,66,131)(51,113,67,132)(52,114,68,133)(53,115,69,127)(54,116,70,128)(55,117,64,129)(56,118,65,130)(57,107,76,123)(58,108,77,124)(59,109,71,125)(60,110,72,126)(61,111,73,120)(62,112,74,121)(63,106,75,122)(134,215,150,203)(135,216,151,197)(136,217,152,198)(137,211,153,199)(138,212,154,200)(139,213,148,201)(140,214,149,202)(141,207,160,191)(142,208,161,192)(143,209,155,193)(144,210,156,194)(145,204,157,195)(146,205,158,196)(147,206,159,190), (1,147,35,150)(2,141,29,151)(3,142,30,152)(4,143,31,153)(5,144,32,154)(6,145,33,148)(7,146,34,149)(8,118,18,121)(9,119,19,122)(10,113,20,123)(11,114,21,124)(12,115,15,125)(13,116,16,126)(14,117,17,120)(22,129,223,111)(23,130,224,112)(24,131,218,106)(25,132,219,107)(26,133,220,108)(27,127,221,109)(28,128,222,110)(36,156,44,138)(37,157,45,139)(38,158,46,140)(39,159,47,134)(40,160,48,135)(41,161,49,136)(42,155,43,137)(50,187,75,162)(51,188,76,163)(52,189,77,164)(53,183,71,165)(54,184,72,166)(55,185,73,167)(56,186,74,168)(57,179,67,169)(58,180,68,170)(59,181,69,171)(60,182,70,172)(61,176,64,173)(62,177,65,174)(63,178,66,175)(78,215,103,190)(79,216,104,191)(80,217,105,192)(81,211,99,193)(82,212,100,194)(83,213,101,195)(84,214,102,196)(85,207,95,197)(86,208,96,198)(87,209,97,199)(88,210,98,200)(89,204,92,201)(90,205,93,202)(91,206,94,203), (1,150)(2,151)(3,152)(4,153)(5,154)(6,148)(7,149)(8,118)(9,119)(10,113)(11,114)(12,115)(13,116)(14,117)(15,125)(16,126)(17,120)(18,121)(19,122)(20,123)(21,124)(22,129)(23,130)(24,131)(25,132)(26,133)(27,127)(28,128)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,138)(37,139)(38,140)(39,134)(40,135)(41,136)(42,137)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(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)], [(1,75,39,63),(2,76,40,57),(3,77,41,58),(4,71,42,59),(5,72,36,60),(6,73,37,61),(7,74,38,62),(8,196,23,205),(9,190,24,206),(10,191,25,207),(11,192,26,208),(12,193,27,209),(13,194,28,210),(14,195,22,204),(15,211,221,199),(16,212,222,200),(17,213,223,201),(18,214,224,202),(19,215,218,203),(20,216,219,197),(21,217,220,198),(29,51,48,67),(30,52,49,68),(31,53,43,69),(32,54,44,70),(33,55,45,64),(34,56,46,65),(35,50,47,66),(78,131,94,119),(79,132,95,113),(80,133,96,114),(81,127,97,115),(82,128,98,116),(83,129,92,117),(84,130,93,118),(85,123,104,107),(86,124,105,108),(87,125,99,109),(88,126,100,110),(89,120,101,111),(90,121,102,112),(91,122,103,106),(134,175,150,187),(135,169,151,188),(136,170,152,189),(137,171,153,183),(138,172,154,184),(139,173,148,185),(140,174,149,186),(141,163,160,179),(142,164,161,180),(143,165,155,181),(144,166,156,182),(145,167,157,176),(146,168,158,177),(147,162,159,178)], [(1,103,39,91),(2,104,40,85),(3,105,41,86),(4,99,42,87),(5,100,36,88),(6,101,37,89),(7,102,38,90),(8,168,23,177),(9,162,24,178),(10,163,25,179),(11,164,26,180),(12,165,27,181),(13,166,28,182),(14,167,22,176),(15,183,221,171),(16,184,222,172),(17,185,223,173),(18,186,224,174),(19,187,218,175),(20,188,219,169),(21,189,220,170),(29,79,48,95),(30,80,49,96),(31,81,43,97),(32,82,44,98),(33,83,45,92),(34,84,46,93),(35,78,47,94),(50,119,66,131),(51,113,67,132),(52,114,68,133),(53,115,69,127),(54,116,70,128),(55,117,64,129),(56,118,65,130),(57,107,76,123),(58,108,77,124),(59,109,71,125),(60,110,72,126),(61,111,73,120),(62,112,74,121),(63,106,75,122),(134,215,150,203),(135,216,151,197),(136,217,152,198),(137,211,153,199),(138,212,154,200),(139,213,148,201),(140,214,149,202),(141,207,160,191),(142,208,161,192),(143,209,155,193),(144,210,156,194),(145,204,157,195),(146,205,158,196),(147,206,159,190)], [(1,147,35,150),(2,141,29,151),(3,142,30,152),(4,143,31,153),(5,144,32,154),(6,145,33,148),(7,146,34,149),(8,118,18,121),(9,119,19,122),(10,113,20,123),(11,114,21,124),(12,115,15,125),(13,116,16,126),(14,117,17,120),(22,129,223,111),(23,130,224,112),(24,131,218,106),(25,132,219,107),(26,133,220,108),(27,127,221,109),(28,128,222,110),(36,156,44,138),(37,157,45,139),(38,158,46,140),(39,159,47,134),(40,160,48,135),(41,161,49,136),(42,155,43,137),(50,187,75,162),(51,188,76,163),(52,189,77,164),(53,183,71,165),(54,184,72,166),(55,185,73,167),(56,186,74,168),(57,179,67,169),(58,180,68,170),(59,181,69,171),(60,182,70,172),(61,176,64,173),(62,177,65,174),(63,178,66,175),(78,215,103,190),(79,216,104,191),(80,217,105,192),(81,211,99,193),(82,212,100,194),(83,213,101,195),(84,214,102,196),(85,207,95,197),(86,208,96,198),(87,209,97,199),(88,210,98,200),(89,204,92,201),(90,205,93,202),(91,206,94,203)], [(1,150),(2,151),(3,152),(4,153),(5,154),(6,148),(7,149),(8,118),(9,119),(10,113),(11,114),(12,115),(13,116),(14,117),(15,125),(16,126),(17,120),(18,121),(19,122),(20,123),(21,124),(22,129),(23,130),(24,131),(25,132),(26,133),(27,127),(28,128),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,138),(37,139),(38,140),(39,134),(40,135),(41,136),(42,137),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,161),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀