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G = (C22×D7).Q8  order 448 = 26·7

1st non-split extension by C22×D7 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C22×D7).1Q8, C22.43(Q8×D7), C14.8(C4⋊D4), (C2×Dic7).14D4, (C22×C4).21D14, C22.160(D4×D7), C2.C42⋊5D7, C7⋊1(C23.Q8), C2.7(C42⋊2D7), C14.27(C22⋊Q8), C2.11(D14⋊D4), (C23×D7).8C22, C14.2(C42⋊2C2), C2.11(D14⋊Q8), C22.93(C4○D28), (C22×C28).20C22, C23.364(C22×D7), (C22×C14).301C23, (C22×Dic7).23C22, (C2×Dic7⋊C4)⋊4C2, (C2×D14⋊C4).9C2, (C2×C14).70(C2×Q8), (C2×C14).208(C2×D4), (C2×C14).61(C4○D4), (C7×C2.C42)⋊2C2, SmallGroup(448,210)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C14 — (C22×D7).Q8
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — C2×D14⋊C4 — (C22×D7).Q8
Lower central
C7 — C22×C14 — (C22×D7).Q8
Upper central
C1 — C23 — C2.C42

Generators and relations for (C22×D7).Q8
 G = < a,b,c,d,e,f | a2=b2=c7=d2=e4=1, f2=ae2, ab=ba, ac=ca, ede-1=ad=da, ae=ea, af=fa, bc=cb, fdf-1=bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, fef-1=be-1 >

Subgroups: 1020 in 186 conjugacy classes, 59 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, C28, D14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C22×D7, C22×D7, C22×C14, C23.Q8, Dic7⋊C4, D14⋊C4, C22×Dic7, C22×C28, C23×D7, C7×C2.C42, C2×Dic7⋊C4, C2×D14⋊C4, (C22×D7).Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C4⋊D4, C22⋊Q8, C42⋊2C2, C22×D7, C23.Q8, C4○D28, D4×D7, Q8×D7, C42⋊2D7, D14⋊D4, D14⋊Q8, (C22×D7).Q8

Smallest permutation representation of (C22×D7).Q8
►On 224 points
Generators in S224
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(28 35)(57 106)(58 107)(59 108)(60 109)(61 110)(62 111)(63 112)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 155)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(127 148)(128 149)(129 150)(130 151)(131 152)(132 153)(133 154)(134 141)(135 142)(136 143)(137 144)(138 145)(139 146)(140 147)(169 218)(170 219)(171 220)(172 221)(173 222)(174 223)(175 224)(176 211)(177 212)(178 213)(179 214)(180 215)(181 216)(182 217)(183 204)(184 205)(185 206)(186 207)(187 208)(188 209)(189 210)(190 197)(191 198)(192 199)(193 200)(194 201)(195 202)(196 203)

(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 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 105)(72 104)(73 103)(74 102)(75 101)(76 100)(77 99)(78 112)(79 111)(80 110)(81 109)(82 108)(83 107)(84 106)(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)(169 217)(170 216)(171 215)(172 214)(173 213)(174 212)(175 211)(176 224)(177 223)(178 222)(179 221)(180 220)(181 219)(182 218)(183 203)(184 202)(185 201)(186 200)(187 199)(188 198)(189 197)(190 210)(191 209)(192 208)(193 207)(194 206)(195 205)(196 204)

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

(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)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,211)(177,212)(178,213)(179,214)(180,215)(181,216)(182,217)(183,204)(184,205)(185,206)(186,207)(187,208)(188,209)(189,210)(190,197)(191,198)(192,199)(193,200)(194,201)(195,202)(196,203), (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)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,105)(72,104)(73,103)(74,102)(75,101)(76,100)(77,99)(78,112)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(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)(169,217)(170,216)(171,215)(172,214)(173,213)(174,212)(175,211)(176,224)(177,223)(178,222)(179,221)(180,220)(181,219)(182,218)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,210)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204), (1,78,22,57)(2,79,23,58)(3,80,24,59)(4,81,25,60)(5,82,26,61)(6,83,27,62)(7,84,28,63)(8,71,15,64)(9,72,16,65)(10,73,17,66)(11,74,18,67)(12,75,19,68)(13,76,20,69)(14,77,21,70)(29,106,50,85)(30,107,51,86)(31,108,52,87)(32,109,53,88)(33,110,54,89)(34,111,55,90)(35,112,56,91)(36,99,43,92)(37,100,44,93)(38,101,45,94)(39,102,46,95)(40,103,47,96)(41,104,48,97)(42,105,49,98)(113,183,134,176)(114,184,135,177)(115,185,136,178)(116,186,137,179)(117,187,138,180)(118,188,139,181)(119,189,140,182)(120,190,127,169)(121,191,128,170)(122,192,129,171)(123,193,130,172)(124,194,131,173)(125,195,132,174)(126,196,133,175)(141,211,162,204)(142,212,163,205)(143,213,164,206)(144,214,165,207)(145,215,166,208)(146,216,167,209)(147,217,168,210)(148,218,155,197)(149,219,156,198)(150,220,157,199)(151,221,158,200)(152,222,159,201)(153,223,160,202)(154,224,161,203), (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( (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34)(28,35)(57,106)(58,107)(59,108)(60,109)(61,110)(62,111)(63,112)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,211)(177,212)(178,213)(179,214)(180,215)(181,216)(182,217)(183,204)(184,205)(185,206)(186,207)(187,208)(188,209)(189,210)(190,197)(191,198)(192,199)(193,200)(194,201)(195,202)(196,203), (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)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,105)(72,104)(73,103)(74,102)(75,101)(76,100)(77,99)(78,112)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(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)(169,217)(170,216)(171,215)(172,214)(173,213)(174,212)(175,211)(176,224)(177,223)(178,222)(179,221)(180,220)(181,219)(182,218)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,210)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204), (1,78,22,57)(2,79,23,58)(3,80,24,59)(4,81,25,60)(5,82,26,61)(6,83,27,62)(7,84,28,63)(8,71,15,64)(9,72,16,65)(10,73,17,66)(11,74,18,67)(12,75,19,68)(13,76,20,69)(14,77,21,70)(29,106,50,85)(30,107,51,86)(31,108,52,87)(32,109,53,88)(33,110,54,89)(34,111,55,90)(35,112,56,91)(36,99,43,92)(37,100,44,93)(38,101,45,94)(39,102,46,95)(40,103,47,96)(41,104,48,97)(42,105,49,98)(113,183,134,176)(114,184,135,177)(115,185,136,178)(116,186,137,179)(117,187,138,180)(118,188,139,181)(119,189,140,182)(120,190,127,169)(121,191,128,170)(122,192,129,171)(123,193,130,172)(124,194,131,173)(125,195,132,174)(126,196,133,175)(141,211,162,204)(142,212,163,205)(143,213,164,206)(144,214,165,207)(145,215,166,208)(146,216,167,209)(147,217,168,210)(148,218,155,197)(149,219,156,198)(150,220,157,199)(151,221,158,200)(152,222,159,201)(153,223,160,202)(154,224,161,203), (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([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34),(28,35),(57,106),(58,107),(59,108),(60,109),(61,110),(62,111),(63,112),(64,99),(65,100),(66,101),(67,102),(68,103),(69,104),(70,105),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91),(113,162),(114,163),(115,164),(116,165),(117,166),(118,167),(119,168),(120,155),(121,156),(122,157),(123,158),(124,159),(125,160),(126,161),(127,148),(128,149),(129,150),(130,151),(131,152),(132,153),(133,154),(134,141),(135,142),(136,143),(137,144),(138,145),(139,146),(140,147),(169,218),(170,219),(171,220),(172,221),(173,222),(174,223),(175,224),(176,211),(177,212),(178,213),(179,214),(180,215),(181,216),(182,217),(183,204),(184,205),(185,206),(186,207),(187,208),(188,209),(189,210),(190,197),(191,198),(192,199),(193,200),(194,201),(195,202),(196,203)], [(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),(57,91),(58,90),(59,89),(60,88),(61,87),(62,86),(63,85),(64,98),(65,97),(66,96),(67,95),(68,94),(69,93),(70,92),(71,105),(72,104),(73,103),(74,102),(75,101),(76,100),(77,99),(78,112),(79,111),(80,110),(81,109),(82,108),(83,107),(84,106),(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),(169,217),(170,216),(171,215),(172,214),(173,213),(174,212),(175,211),(176,224),(177,223),(178,222),(179,221),(180,220),(181,219),(182,218),(183,203),(184,202),(185,201),(186,200),(187,199),(188,198),(189,197),(190,210),(191,209),(192,208),(193,207),(194,206),(195,205),(196,204)], [(1,78,22,57),(2,79,23,58),(3,80,24,59),(4,81,25,60),(5,82,26,61),(6,83,27,62),(7,84,28,63),(8,71,15,64),(9,72,16,65),(10,73,17,66),(11,74,18,67),(12,75,19,68),(13,76,20,69),(14,77,21,70),(29,106,50,85),(30,107,51,86),(31,108,52,87),(32,109,53,88),(33,110,54,89),(34,111,55,90),(35,112,56,91),(36,99,43,92),(37,100,44,93),(38,101,45,94),(39,102,46,95),(40,103,47,96),(41,104,48,97),(42,105,49,98),(113,183,134,176),(114,184,135,177),(115,185,136,178),(116,186,137,179),(117,187,138,180),(118,188,139,181),(119,189,140,182),(120,190,127,169),(121,191,128,170),(122,192,129,171),(123,193,130,172),(124,194,131,173),(125,195,132,174),(126,196,133,175),(141,211,162,204),(142,212,163,205),(143,213,164,206),(144,214,165,207),(145,215,166,208),(146,216,167,209),(147,217,168,210),(148,218,155,197),(149,219,156,198),(150,220,157,199),(151,221,158,200),(152,222,159,201),(153,223,160,202),(154,224,161,203)], [(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···2G2H2I4A···4F4G···4L7A7B7C14A···14U28A···28AJ
order12···2224···44···477714···1428···28
size11···128284···428···282222···24···4

82 irreducible representations

dim111122222244
type+++++-+++-
imageC1C2C2C2D4Q8D7C4○D4D14C4○D28D4×D7Q8×D7
kernel(C22×D7).Q8C7×C2.C42C2×Dic7⋊C4C2×D14⋊C4C2×Dic7C22×D7C2.C42C2×C14C22×C4C22C22C22
# reps1133623693693

Matrix representation of (C22×D7).Q8 ►in GL6(𝔽29)

2800000
0280000
0028000
0002800
0000280
0000028
,
100000
010000
0028000
0002800
0000280
0000028
,
3280000
100000
001000
000100
00001928
00002028
,
3280000
8260000
001000
00162800
00002211
0000227
,
24130000
1650000
0082800
0052100
00002124
0000138
,
1700000
0170000
0082800
0072100
00002027
0000119

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[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],[3,1,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,20,0,0,0,0,28,28],[3,8,0,0,0,0,28,26,0,0,0,0,0,0,1,16,0,0,0,0,0,28,0,0,0,0,0,0,22,22,0,0,0,0,11,7],[24,16,0,0,0,0,13,5,0,0,0,0,0,0,8,5,0,0,0,0,28,21,0,0,0,0,0,0,21,13,0,0,0,0,24,8],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,8,7,0,0,0,0,28,21,0,0,0,0,0,0,20,11,0,0,0,0,27,9] >;
 

(C22×D7).Q8 in GAP, Magma, Sage, TeX 

(C_2^2\times D_7).Q_8
 
% in TeX
 
G:=Group("(C2^2xD7).Q8");
 
// GroupNames label
 
G:=SmallGroup(448,210);
 
// by ID
 
G=gap.SmallGroup(448,210);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,64,1262,387,268,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^7=d^2=e^4=1,f^2=a*e^2,a*b=b*a,a*c=c*a,e*d*e^-1=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,f*e*f^-1=b*e^-1>;
 
// generators/relations
 

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