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## G = C7×(C22×C8)⋊C2order 448 = 26·7

### Direct product of C7 and (C22×C8)⋊C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×(C22×C8)⋊C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C2×C56 — C7×C22⋊C8 — C7×(C22×C8)⋊C2
 Lower central C1 — C22 — C7×(C22×C8)⋊C2
 Upper central C1 — C2×C28 — C7×(C22×C8)⋊C2

Generators and relations for C7×(C22×C8)⋊C2
G = < a,b,c,d,e | a7=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bd4, ede=cd=dc, ce=ec >

Subgroups: 242 in 158 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C14, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, (C22×C8)⋊C2, C2×C56, C2×C56, C7×M4(2), C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C22⋊C8, C22×C56, C14×M4(2), C14×C4○D4, C7×(C22×C8)⋊C2
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C2×C22⋊C4, C8○D4, C2×C28, C7×D4, C22×C14, (C22×C8)⋊C2, C7×C22⋊C4, C22×C28, D4×C14, C14×C22⋊C4, C7×C8○D4, C7×(C22×C8)⋊C2

Smallest permutation representation of C7×(C22×C8)⋊C2
On 224 points
Generators in S224
(1 147 35 139 27 131 19)(2 148 36 140 28 132 20)(3 149 37 141 29 133 21)(4 150 38 142 30 134 22)(5 151 39 143 31 135 23)(6 152 40 144 32 136 24)(7 145 33 137 25 129 17)(8 146 34 138 26 130 18)(9 64 168 56 160 48 126)(10 57 161 49 153 41 127)(11 58 162 50 154 42 128)(12 59 163 51 155 43 121)(13 60 164 52 156 44 122)(14 61 165 53 157 45 123)(15 62 166 54 158 46 124)(16 63 167 55 159 47 125)(65 169 193 81 185 73 177)(66 170 194 82 186 74 178)(67 171 195 83 187 75 179)(68 172 196 84 188 76 180)(69 173 197 85 189 77 181)(70 174 198 86 190 78 182)(71 175 199 87 191 79 183)(72 176 200 88 192 80 184)(89 117 217 105 209 97 201)(90 118 218 106 210 98 202)(91 119 219 107 211 99 203)(92 120 220 108 212 100 204)(93 113 221 109 213 101 205)(94 114 222 110 214 102 206)(95 115 223 111 215 103 207)(96 116 224 112 216 104 208)
(1 175)(2 176)(3 169)(4 170)(5 171)(6 172)(7 173)(8 174)(9 224)(10 217)(11 218)(12 219)(13 220)(14 221)(15 222)(16 223)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(25 77)(26 78)(27 79)(28 80)(29 73)(30 74)(31 75)(32 76)(33 85)(34 86)(35 87)(36 88)(37 81)(38 82)(39 83)(40 84)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(113 123)(114 124)(115 125)(116 126)(117 127)(118 128)(119 121)(120 122)(129 181)(130 182)(131 183)(132 184)(133 177)(134 178)(135 179)(136 180)(137 189)(138 190)(139 191)(140 192)(141 185)(142 186)(143 187)(144 188)(145 197)(146 198)(147 199)(148 200)(149 193)(150 194)(151 195)(152 196)(153 201)(154 202)(155 203)(156 204)(157 205)(158 206)(159 207)(160 208)(161 209)(162 210)(163 211)(164 212)(165 213)(166 214)(167 215)(168 216)
(1 175)(2 176)(3 169)(4 170)(5 171)(6 172)(7 173)(8 174)(9 220)(10 221)(11 222)(12 223)(13 224)(14 217)(15 218)(16 219)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(25 77)(26 78)(27 79)(28 80)(29 73)(30 74)(31 75)(32 76)(33 85)(34 86)(35 87)(36 88)(37 81)(38 82)(39 83)(40 84)(41 93)(42 94)(43 95)(44 96)(45 89)(46 90)(47 91)(48 92)(49 101)(50 102)(51 103)(52 104)(53 97)(54 98)(55 99)(56 100)(57 109)(58 110)(59 111)(60 112)(61 105)(62 106)(63 107)(64 108)(113 127)(114 128)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(129 181)(130 182)(131 183)(132 184)(133 177)(134 178)(135 179)(136 180)(137 189)(138 190)(139 191)(140 192)(141 185)(142 186)(143 187)(144 188)(145 197)(146 198)(147 199)(148 200)(149 193)(150 194)(151 195)(152 196)(153 205)(154 206)(155 207)(156 208)(157 201)(158 202)(159 203)(160 204)(161 213)(162 214)(163 215)(164 216)(165 209)(166 210)(167 211)(168 212)
(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 43)(2 96)(3 45)(4 90)(5 47)(6 92)(7 41)(8 94)(9 84)(10 33)(11 86)(12 35)(13 88)(14 37)(15 82)(16 39)(17 153)(18 206)(19 155)(20 208)(21 157)(22 202)(23 159)(24 204)(25 161)(26 214)(27 163)(28 216)(29 165)(30 210)(31 167)(32 212)(34 222)(36 224)(38 218)(40 220)(42 174)(44 176)(46 170)(48 172)(49 129)(50 182)(51 131)(52 184)(53 133)(54 178)(55 135)(56 180)(57 137)(58 190)(59 139)(60 192)(61 141)(62 186)(63 143)(64 188)(65 201)(66 158)(67 203)(68 160)(69 205)(70 154)(71 207)(72 156)(73 209)(74 166)(75 211)(76 168)(77 213)(78 162)(79 215)(80 164)(81 217)(83 219)(85 221)(87 223)(89 169)(91 171)(93 173)(95 175)(97 177)(98 134)(99 179)(100 136)(101 181)(102 130)(103 183)(104 132)(105 185)(106 142)(107 187)(108 144)(109 189)(110 138)(111 191)(112 140)(113 197)(114 146)(115 199)(116 148)(117 193)(118 150)(119 195)(120 152)(121 147)(122 200)(123 149)(124 194)(125 151)(126 196)(127 145)(128 198)

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

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

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

196 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 7A ··· 7F 8A ··· 8H 8I 8J 8K 8L 14A ··· 14R 14S ··· 14AD 14AE ··· 14AP 28A ··· 28X 28Y ··· 28AJ 28AK ··· 28AV 56A ··· 56AV 56AW ··· 56BT order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 ··· 7 8 ··· 8 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 4 1 ··· 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C7 C14 C14 C14 C14 C28 C28 D4 C8○D4 C7×D4 C7×C8○D4 kernel C7×(C22×C8)⋊C2 C7×C22⋊C8 C22×C56 C14×M4(2) C14×C4○D4 D4×C14 Q8×C14 (C22×C8)⋊C2 C22⋊C8 C22×C8 C2×M4(2) C2×C4○D4 C2×D4 C2×Q8 C2×C28 C14 C2×C4 C2 # reps 1 4 1 1 1 6 2 6 24 6 6 6 36 12 4 8 24 48

Matrix representation of C7×(C22×C8)⋊C2 in GL4(𝔽113) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 112 112 0 0 0 1 0 0 0 0 1 0 0 0 0 112
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 18 18 0 0 0 95 0 0 0 0 95 0 0 0 0 95
,
 1 0 0 0 111 112 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[112,0,0,0,112,1,0,0,0,0,1,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[18,0,0,0,18,95,0,0,0,0,95,0,0,0,0,95],[1,111,0,0,0,112,0,0,0,0,0,1,0,0,1,0] >;

C7×(C22×C8)⋊C2 in GAP, Magma, Sage, TeX

C_7\times (C_2^2\times C_8)\rtimes C_2
% in TeX

G:=Group("C7x(C2^2xC8):C2");
// GroupNames label

G:=SmallGroup(448,816);
// by ID

G=gap.SmallGroup(448,816);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,2403,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b*d^4,e*d*e=c*d=d*c,c*e=e*c>;
// generators/relations

׿
×
𝔽