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G = C7×C23.10D4  order 448 = 26·7

Direct product of C7 and C23.10D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×C23.10D4, (C2×C28)⋊25D4, C23.9(C7×D4), C24.9(C2×C14), C14.92C22≀C2, (C22×D4).3C14, (C22×C14).28D4, C22.70(D4×C14), (C23×C14).6C22, C14.138(C4⋊D4), C2.C42⋊11C14, C14.68(C4.4D4), C23.77(C22×C14), (C22×C14).458C23, (C22×C28).402C22, C14.90(C22.D4), (C2×C4)⋊3(C7×D4), (C2×C4⋊C4)⋊5C14, (C14×C4⋊C4)⋊32C2, (D4×C2×C14).15C2, C2.7(C7×C4⋊D4), (C14×C22⋊C4)⋊8C2, (C2×C22⋊C4)⋊7C14, C2.6(C7×C22≀C2), C2.6(C7×C4.4D4), (C2×C14).610(C2×D4), (C22×C4).6(C2×C14), C22.37(C7×C4○D4), (C2×C14).218(C4○D4), C2.6(C7×C22.D4), (C7×C2.C42)⋊27C2, SmallGroup(448,802)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C7×C23.10D4
Chief series
C1 — C2 — C22 — C23 — C22×C14 — C23×C14 — D4×C2×C14 — C7×C23.10D4
Lower central
C1 — C23 — C7×C23.10D4
Upper central
C1 — C22×C14 — C7×C23.10D4

Generators and relations for C7×C23.10D4
 G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, ebe-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce-1 >

Subgroups: 458 in 238 conjugacy classes, 78 normal (30 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C23.10D4, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, C23×C14, C7×C2.C42, C14×C22⋊C4, C14×C4⋊C4, D4×C2×C14, C7×C23.10D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C7×D4, C22×C14, C23.10D4, D4×C14, C7×C4○D4, C7×C22≀C2, C7×C4⋊D4, C7×C22.D4, C7×C4.4D4, C7×C23.10D4

Smallest permutation representation of C7×C23.10D4
►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)

(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 215)(16 216)(17 217)(18 211)(19 212)(20 213)(21 214)(29 218)(30 219)(31 220)(32 221)(33 222)(34 223)(35 224)(36 45)(37 46)(38 47)(39 48)(40 49)(41 43)(42 44)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(155 199)(156 200)(157 201)(158 202)(159 203)(160 197)(161 198)(162 181)(163 182)(164 176)(165 177)(166 178)(167 179)(168 180)(169 205)(170 206)(171 207)(172 208)(173 209)(174 210)(175 204)(183 195)(184 196)(185 190)(186 191)(187 192)(188 193)(189 194)

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

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

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

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

154 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4J7A···7F14A···14AP14AQ···14BN28A···28BH
order12···222224···47···714···1414···1428···28
size11···144444···41···11···14···44···4

154 irreducible representations

dim1111111111222222
type+++++++
imageC1C2C2C2C2C7C14C14C14C14D4D4C4○D4C7×D4C7×D4C7×C4○D4
kernelC7×C23.10D4C7×C2.C42C14×C22⋊C4C14×C4⋊C4D4×C2×C14C23.10D4C2.C42C2×C22⋊C4C2×C4⋊C4C22×D4C2×C28C22×C14C2×C14C2×C4C23C22
# reps11411662466446242436

Matrix representation of C7×C23.10D4 ►in GL6(𝔽29)

2000000
0200000
001000
000100
0000240
0000024
,
100000
0280000
001000
0002800
000010
00002828
,
100000
010000
001000
000100
0000280
0000028
,
2800000
0280000
0028000
0002800
000010
000001
,
0280000
100000
000100
001000
00001224
0000017
,
100000
0280000
001000
000100
000012
0000028

G:=sub<GL(6,GF(29))| [20,0,0,0,0,0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,0,0,0,0,0,0,24],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,28,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,24,17],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,28] >;
 

C7×C23.10D4 in GAP, Magma, Sage, TeX 

C_7\times C_2^3._{10}D_4
 
% in TeX
 
G:=Group("C7xC2^3.10D4");
 
// GroupNames label
 
G:=SmallGroup(448,802);
 
// by ID
 
G=gap.SmallGroup(448,802);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,2438,2403,310]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f=b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*e^-1>;
 
// generators/relations
 

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