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G = C4.Q8⋊D7  order 448 = 26·7

9th semidirect product of C4.Q8 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4.Q8⋊9D7, D14⋊C8⋊31C2, C4⋊C4.40D14, C4⋊D28.6C2, (C2×C8).139D14, C2.D56⋊32C2, C14.D8⋊17C2, C28.31(C4○D4), C4.74(C4○D28), C14.56(C4○D8), C28.Q8⋊16C2, (C22×D7).25D4, C22.218(D4×D7), C2.23(D56⋊C2), C14.71(C8⋊C22), (C2×C28).282C23, (C2×C56).286C22, C4.26(Q8⋊2D7), (C2×Dic7).163D4, (C2×D28).76C22, C7⋊4(C23.19D4), C4⋊Dic7.112C22, C2.23(SD16⋊3D7), C2.13(D14.5D4), C14.43(C22.D4), C4⋊C4⋊7D7⋊6C2, (C7×C4.Q8)⋊17C2, (C2×C7⋊C8).59C22, (C2×C4×D7).34C22, (C2×C14).287(C2×D4), (C7×C4⋊C4).75C22, (C2×C4).385(C22×D7), SmallGroup(448,400)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C28 — C4.Q8⋊D7
Chief series
C1 — C7 — C14 — C2×C14 — C2×C28 — C2×C4×D7 — C4⋊C4⋊7D7 — C4.Q8⋊D7
Lower central
C7 — C14 — C2×C28 — C4.Q8⋊D7
Upper central
C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C4.Q8⋊D7
 G = < a,b,c,d,e | a4=d7=e2=1, b4=a2, c2=a-1b2, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b3, bd=db, ebe=a-1b3, cd=dc, ece=a2c, ede=d-1 >

Subgroups: 684 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C7⋊C8, C56, C4×D7, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C23.19D4, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, C2×D28, C28.Q8, C14.D8, D14⋊C8, C2.D56, C7×C4.Q8, C4⋊C4⋊7D7, C4⋊D28, C4.Q8⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C4○D8, C8⋊C22, C22×D7, C23.19D4, C4○D28, D4×D7, Q8⋊2D7, D14.5D4, D56⋊C2, SD16⋊3D7, C4.Q8⋊D7

Smallest permutation representation of C4.Q8⋊D7
►On 224 points
Generators in S224
(1 53 5 49)(2 54 6 50)(3 55 7 51)(4 56 8 52)(9 196 13 200)(10 197 14 193)(11 198 15 194)(12 199 16 195)(17 164 21 168)(18 165 22 161)(19 166 23 162)(20 167 24 163)(25 181 29 177)(26 182 30 178)(27 183 31 179)(28 184 32 180)(33 113 37 117)(34 114 38 118)(35 115 39 119)(36 116 40 120)(41 125 45 121)(42 126 46 122)(43 127 47 123)(44 128 48 124)(57 147 61 151)(58 148 62 152)(59 149 63 145)(60 150 64 146)(65 201 69 205)(66 202 70 206)(67 203 71 207)(68 204 72 208)(73 211 77 215)(74 212 78 216)(75 213 79 209)(76 214 80 210)(81 109 85 105)(82 110 86 106)(83 111 87 107)(84 112 88 108)(89 135 93 131)(90 136 94 132)(91 129 95 133)(92 130 96 134)(97 174 101 170)(98 175 102 171)(99 176 103 172)(100 169 104 173)(137 220 141 224)(138 221 142 217)(139 222 143 218)(140 223 144 219)(153 188 157 192)(154 189 158 185)(155 190 159 186)(156 191 160 187)

(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)

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

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

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

61 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I7A7B7C8A8B8C8D14A···14I28A···28F28G···28R56A···56L
order122222444444444777888814···1428···2828···2856···56
size1111285622448141428282224428282···24···48···84···4

61 irreducible representations

dim111111112222222244444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D7C4○D4D14D14C4○D8C4○D28C8⋊C22Q8⋊2D7D4×D7D56⋊C2SD16⋊3D7
kernelC4.Q8⋊D7C28.Q8C14.D8D14⋊C8C2.D56C7×C4.Q8C4⋊C4⋊7D7C4⋊D28C2×Dic7C22×D7C4.Q8C28C4⋊C4C2×C8C14C4C14C4C22C2C2
# reps1111111111346341213366

Matrix representation of C4.Q8⋊D7 ►in GL6(𝔽113)

100000
010000
001000
000100
0000980
000011215
,
010000
11200000
001000
000100
0000950
00001344
,
9800000
0150000
001000
000100
000098111
000011215
,
100000
010000
0088100
00533400
000010
000001
,
100000
01120000
0034900
00607900
000010
000098112

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,98,112,0,0,0,0,0,15],[0,112,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,95,13,0,0,0,0,0,44],[98,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,98,112,0,0,0,0,111,15],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,88,53,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,112,0,0,0,0,0,0,34,60,0,0,0,0,9,79,0,0,0,0,0,0,1,98,0,0,0,0,0,112] >;
 

C4.Q8⋊D7 in GAP, Magma, Sage, TeX 

C_4.Q_8\rtimes D_7
 
% in TeX
 
G:=Group("C4.Q8:D7");
 
// GroupNames label
 
G:=SmallGroup(448,400);
 
// by ID
 
G=gap.SmallGroup(448,400);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,64,254,219,100,851,102,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=d^7=e^2=1,b^4=a^2,c^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^3,b*d=d*b,e*b*e=a^-1*b^3,c*d=d*c,e*c*e=a^2*c,e*d*e=d^-1>;
 
// generators/relations
 

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