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## G = (C22×D7).9D4order 448 = 26·7

### 3rd non-split extension by C22×D7 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C14 — (C22×D7).9D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — C2×D14⋊C4 — (C22×D7).9D4
 Lower central C7 — C22×C14 — (C22×D7).9D4
 Upper central C1 — C23 — C2.C42

Generators and relations for (C22×D7).9D4
G = < a,b,c,d,e,f | a2=b2=c7=d2=e4=1, f2=ba=ab, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fdf-1=bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, ede-1=abd, fef-1=ae-1 >

Subgroups: 956 in 170 conjugacy classes, 55 normal (51 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C23, C23, D7, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, C28, D14, C2×C14, C2.C42, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C22×D7, C22×D7, C22×C14, C23.11D4, Dic7⋊C4, D14⋊C4, C22×Dic7, C22×C28, C23×D7, C14.C42, C7×C2.C42, C2×Dic7⋊C4, C2×D14⋊C4, (C22×D7).9D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C22.D4, C4.4D4, C422C2, C22×D7, C23.11D4, C4○D28, D4×D7, D42D7, Q82D7, C422D7, D14.D4, D14⋊D4, Dic7.D4, D14.5D4, C4⋊C4⋊D7, (C22×D7).9D4

Smallest permutation representation of (C22×D7).9D4
On 224 points
Generators in S224
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(113 155)(114 156)(115 157)(116 158)(117 159)(118 160)(119 161)(120 162)(121 163)(122 164)(123 165)(124 166)(125 167)(126 168)(127 141)(128 142)(129 143)(130 144)(131 145)(132 146)(133 147)(134 148)(135 149)(136 150)(137 151)(138 152)(139 153)(140 154)(169 211)(170 212)(171 213)(172 214)(173 215)(174 216)(175 217)(176 218)(177 219)(178 220)(179 221)(180 222)(181 223)(182 224)(183 197)(184 198)(185 199)(186 200)(187 201)(188 202)(189 203)(190 204)(191 205)(192 206)(193 207)(194 208)(195 209)(196 210)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(57 98)(58 97)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(155 168)(156 167)(157 166)(158 165)(159 164)(160 163)(161 162)(169 224)(170 223)(171 222)(172 221)(173 220)(174 219)(175 218)(176 217)(177 216)(178 215)(179 214)(180 213)(181 212)(182 211)(183 210)(184 209)(185 208)(186 207)(187 206)(188 205)(189 204)(190 203)(191 202)(192 201)(193 200)(194 199)(195 198)(196 197)
(1 218 50 169)(2 219 51 170)(3 220 52 171)(4 221 53 172)(5 222 54 173)(6 223 55 174)(7 224 56 175)(8 211 43 176)(9 212 44 177)(10 213 45 178)(11 214 46 179)(12 215 47 180)(13 216 48 181)(14 217 49 182)(15 204 36 183)(16 205 37 184)(17 206 38 185)(18 207 39 186)(19 208 40 187)(20 209 41 188)(21 210 42 189)(22 197 29 190)(23 198 30 191)(24 199 31 192)(25 200 32 193)(26 201 33 194)(27 202 34 195)(28 203 35 196)(57 148 106 127)(58 149 107 128)(59 150 108 129)(60 151 109 130)(61 152 110 131)(62 153 111 132)(63 154 112 133)(64 141 99 134)(65 142 100 135)(66 143 101 136)(67 144 102 137)(68 145 103 138)(69 146 104 139)(70 147 105 140)(71 162 92 113)(72 163 93 114)(73 164 94 115)(74 165 95 116)(75 166 96 117)(76 167 97 118)(77 168 98 119)(78 155 85 120)(79 156 86 121)(80 157 87 122)(81 158 88 123)(82 159 89 124)(83 160 90 125)(84 161 91 126)
(1 141 29 113)(2 142 30 114)(3 143 31 115)(4 144 32 116)(5 145 33 117)(6 146 34 118)(7 147 35 119)(8 148 36 120)(9 149 37 121)(10 150 38 122)(11 151 39 123)(12 152 40 124)(13 153 41 125)(14 154 42 126)(15 155 43 127)(16 156 44 128)(17 157 45 129)(18 158 46 130)(19 159 47 131)(20 160 48 132)(21 161 49 133)(22 162 50 134)(23 163 51 135)(24 164 52 136)(25 165 53 137)(26 166 54 138)(27 167 55 139)(28 168 56 140)(57 197 85 169)(58 198 86 170)(59 199 87 171)(60 200 88 172)(61 201 89 173)(62 202 90 174)(63 203 91 175)(64 204 92 176)(65 205 93 177)(66 206 94 178)(67 207 95 179)(68 208 96 180)(69 209 97 181)(70 210 98 182)(71 211 99 183)(72 212 100 184)(73 213 101 185)(74 214 102 186)(75 215 103 187)(76 216 104 188)(77 217 105 189)(78 218 106 190)(79 219 107 191)(80 220 108 192)(81 221 109 193)(82 222 110 194)(83 223 111 195)(84 224 112 196)

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

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

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

82 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4F 4G ··· 4L 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 28 28 4 ··· 4 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 D4 D4 D7 C4○D4 D14 C4○D28 D4×D7 D4⋊2D7 Q8⋊2D7 kernel (C22×D7).9D4 C14.C42 C7×C2.C42 C2×Dic7⋊C4 C2×D14⋊C4 C2×Dic7 C22×D7 C2.C42 C2×C14 C22×C4 C22 C22 C22 C22 # reps 1 2 1 1 3 2 2 3 10 9 36 6 3 3

Matrix representation of (C22×D7).9D4 in GL6(𝔽29)

 28 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 0 1
,
 1 0 0 0 0 0 0 1 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 28
,
 21 28 0 0 0 0 2 11 0 0 0 0 0 0 18 28 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 20 21 0 0 0 0 10 9 0 0 0 0 0 0 11 1 0 0 0 0 25 18 0 0 0 0 0 0 1 0 0 0 0 0 17 28
,
 26 18 0 0 0 0 22 3 0 0 0 0 0 0 18 27 0 0 0 0 2 11 0 0 0 0 0 0 1 5 0 0 0 0 17 28
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 13 5 0 0 0 0 24 16 0 0 0 0 0 0 17 27 0 0 0 0 0 12

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[21,2,0,0,0,0,28,11,0,0,0,0,0,0,18,1,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,10,0,0,0,0,21,9,0,0,0,0,0,0,11,25,0,0,0,0,1,18,0,0,0,0,0,0,1,17,0,0,0,0,0,28],[26,22,0,0,0,0,18,3,0,0,0,0,0,0,18,2,0,0,0,0,27,11,0,0,0,0,0,0,1,17,0,0,0,0,5,28],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,13,24,0,0,0,0,5,16,0,0,0,0,0,0,17,0,0,0,0,0,27,12] >;

(C22×D7).9D4 in GAP, Magma, Sage, TeX

(C_2^2\times D_7)._9D_4
% in TeX

G:=Group("(C2^2xD7).9D4");
// GroupNames label

G:=SmallGroup(448,209);
// by ID

G=gap.SmallGroup(448,209);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,64,590,387,100,18822]);
// Polycyclic

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

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