metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊C8⋊5D4, C7⋊4(C8⋊D4), C4⋊C4.62D14, C4⋊D4.7D7, (C2×C28).74D4, C4.173(D4×D7), (C2×D4).42D14, C28.151(C2×D4), D4⋊Dic7⋊18C2, C28.Q8⋊37C2, C14.Q16⋊36C2, (C22×C14).88D4, C28.186(C4○D4), C28.48D4⋊25C2, C4.62(D4⋊2D7), C14.96(C4⋊D4), C14.92(C8⋊C22), (C2×C28).361C23, (D4×C14).58C22, (C22×C4).123D14, C23.25(C7⋊D4), C4⋊Dic7.145C22, C2.17(Dic7⋊D4), C2.13(D4.9D14), C2.13(D4.D14), C14.115(C8.C22), (C22×C28).165C22, (C2×Dic14).104C22, (C2×D4.D7)⋊11C2, (C7×C4⋊D4).6C2, (C2×C14).492(C2×D4), (C2×C4).52(C7⋊D4), (C2×C7⋊C8).111C22, (C2×C4.Dic7)⋊12C2, (C7×C4⋊C4).109C22, (C2×C4).461(C22×D7), C22.167(C2×C7⋊D4), SmallGroup(448,576)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊C8⋊5D4
G = < a,b,c,d | a7=b8=c4=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, dbd=b5, dcd=c-1 >
Subgroups: 492 in 120 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C7⋊C8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C8⋊D4, C2×C7⋊C8, C4.Dic7, Dic7⋊C4, C4⋊Dic7, D4.D7, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C22×C28, D4×C14, D4×C14, C28.Q8, C14.Q16, D4⋊Dic7, C2×C4.Dic7, C28.48D4, C2×D4.D7, C7×C4⋊D4, C7⋊C8⋊5D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, C8.C22, C7⋊D4, C22×D7, C8⋊D4, D4×D7, D4⋊2D7, C2×C7⋊D4, D4.D14, Dic7⋊D4, D4.9D14, C7⋊C8⋊5D4
►On 224 points
Generators in S224
(1 184 26 77 115 39 217)(2 218 40 116 78 27 177)(3 178 28 79 117 33 219)(4 220 34 118 80 29 179)(5 180 30 73 119 35 221)(6 222 36 120 74 31 181)(7 182 32 75 113 37 223)(8 224 38 114 76 25 183)(9 195 160 24 174 57 89)(10 90 58 175 17 153 196)(11 197 154 18 176 59 91)(12 92 60 169 19 155 198)(13 199 156 20 170 61 93)(14 94 62 171 21 157 200)(15 193 158 22 172 63 95)(16 96 64 173 23 159 194)(41 84 122 187 212 106 152)(42 145 107 213 188 123 85)(43 86 124 189 214 108 146)(44 147 109 215 190 125 87)(45 88 126 191 216 110 148)(46 149 111 209 192 127 81)(47 82 128 185 210 112 150)(48 151 105 211 186 121 83)(49 207 135 104 161 66 140)(50 141 67 162 97 136 208)(51 201 129 98 163 68 142)(52 143 69 164 99 130 202)(53 203 131 100 165 70 144)(54 137 71 166 101 132 204)(55 205 133 102 167 72 138)(56 139 65 168 103 134 206)
(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)
(1 156 43 161)(2 155 44 168)(3 154 45 167)(4 153 46 166)(5 160 47 165)(6 159 48 164)(7 158 41 163)(8 157 42 162)(9 112 131 35)(10 111 132 34)(11 110 133 33)(12 109 134 40)(13 108 135 39)(14 107 136 38)(15 106 129 37)(16 105 130 36)(17 81 71 179)(18 88 72 178)(19 87 65 177)(20 86 66 184)(21 85 67 183)(22 84 68 182)(23 83 69 181)(24 82 70 180)(25 171 123 141)(26 170 124 140)(27 169 125 139)(28 176 126 138)(29 175 127 137)(30 174 128 144)(31 173 121 143)(32 172 122 142)(49 77 61 189)(50 76 62 188)(51 75 63 187)(52 74 64 186)(53 73 57 185)(54 80 58 192)(55 79 59 191)(56 78 60 190)(89 210 203 119)(90 209 204 118)(91 216 205 117)(92 215 206 116)(93 214 207 115)(94 213 208 114)(95 212 201 113)(96 211 202 120)(97 224 200 145)(98 223 193 152)(99 222 194 151)(100 221 195 150)(101 220 196 149)(102 219 197 148)(103 218 198 147)(104 217 199 146)
(2 6)(4 8)(9 131)(10 136)(11 133)(12 130)(13 135)(14 132)(15 129)(16 134)(17 67)(18 72)(19 69)(20 66)(21 71)(22 68)(23 65)(24 70)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(49 61)(50 58)(51 63)(52 60)(53 57)(54 62)(55 59)(56 64)(74 78)(76 80)(81 85)(83 87)(89 203)(90 208)(91 205)(92 202)(93 207)(94 204)(95 201)(96 206)(97 196)(98 193)(99 198)(100 195)(101 200)(102 197)(103 194)(104 199)(105 109)(107 111)(114 118)(116 120)(121 125)(123 127)(137 171)(138 176)(139 173)(140 170)(141 175)(142 172)(143 169)(144 174)(145 149)(147 151)(153 162)(154 167)(155 164)(156 161)(157 166)(158 163)(159 168)(160 165)(177 181)(179 183)(186 190)(188 192)(209 213)(211 215)(218 222)(220 224)
G:=sub<Sym(224)| (1,184,26,77,115,39,217)(2,218,40,116,78,27,177)(3,178,28,79,117,33,219)(4,220,34,118,80,29,179)(5,180,30,73,119,35,221)(6,222,36,120,74,31,181)(7,182,32,75,113,37,223)(8,224,38,114,76,25,183)(9,195,160,24,174,57,89)(10,90,58,175,17,153,196)(11,197,154,18,176,59,91)(12,92,60,169,19,155,198)(13,199,156,20,170,61,93)(14,94,62,171,21,157,200)(15,193,158,22,172,63,95)(16,96,64,173,23,159,194)(41,84,122,187,212,106,152)(42,145,107,213,188,123,85)(43,86,124,189,214,108,146)(44,147,109,215,190,125,87)(45,88,126,191,216,110,148)(46,149,111,209,192,127,81)(47,82,128,185,210,112,150)(48,151,105,211,186,121,83)(49,207,135,104,161,66,140)(50,141,67,162,97,136,208)(51,201,129,98,163,68,142)(52,143,69,164,99,130,202)(53,203,131,100,165,70,144)(54,137,71,166,101,132,204)(55,205,133,102,167,72,138)(56,139,65,168,103,134,206), (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), (1,156,43,161)(2,155,44,168)(3,154,45,167)(4,153,46,166)(5,160,47,165)(6,159,48,164)(7,158,41,163)(8,157,42,162)(9,112,131,35)(10,111,132,34)(11,110,133,33)(12,109,134,40)(13,108,135,39)(14,107,136,38)(15,106,129,37)(16,105,130,36)(17,81,71,179)(18,88,72,178)(19,87,65,177)(20,86,66,184)(21,85,67,183)(22,84,68,182)(23,83,69,181)(24,82,70,180)(25,171,123,141)(26,170,124,140)(27,169,125,139)(28,176,126,138)(29,175,127,137)(30,174,128,144)(31,173,121,143)(32,172,122,142)(49,77,61,189)(50,76,62,188)(51,75,63,187)(52,74,64,186)(53,73,57,185)(54,80,58,192)(55,79,59,191)(56,78,60,190)(89,210,203,119)(90,209,204,118)(91,216,205,117)(92,215,206,116)(93,214,207,115)(94,213,208,114)(95,212,201,113)(96,211,202,120)(97,224,200,145)(98,223,193,152)(99,222,194,151)(100,221,195,150)(101,220,196,149)(102,219,197,148)(103,218,198,147)(104,217,199,146), (2,6)(4,8)(9,131)(10,136)(11,133)(12,130)(13,135)(14,132)(15,129)(16,134)(17,67)(18,72)(19,69)(20,66)(21,71)(22,68)(23,65)(24,70)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(49,61)(50,58)(51,63)(52,60)(53,57)(54,62)(55,59)(56,64)(74,78)(76,80)(81,85)(83,87)(89,203)(90,208)(91,205)(92,202)(93,207)(94,204)(95,201)(96,206)(97,196)(98,193)(99,198)(100,195)(101,200)(102,197)(103,194)(104,199)(105,109)(107,111)(114,118)(116,120)(121,125)(123,127)(137,171)(138,176)(139,173)(140,170)(141,175)(142,172)(143,169)(144,174)(145,149)(147,151)(153,162)(154,167)(155,164)(156,161)(157,166)(158,163)(159,168)(160,165)(177,181)(179,183)(186,190)(188,192)(209,213)(211,215)(218,222)(220,224)>;
G:=Group( (1,184,26,77,115,39,217)(2,218,40,116,78,27,177)(3,178,28,79,117,33,219)(4,220,34,118,80,29,179)(5,180,30,73,119,35,221)(6,222,36,120,74,31,181)(7,182,32,75,113,37,223)(8,224,38,114,76,25,183)(9,195,160,24,174,57,89)(10,90,58,175,17,153,196)(11,197,154,18,176,59,91)(12,92,60,169,19,155,198)(13,199,156,20,170,61,93)(14,94,62,171,21,157,200)(15,193,158,22,172,63,95)(16,96,64,173,23,159,194)(41,84,122,187,212,106,152)(42,145,107,213,188,123,85)(43,86,124,189,214,108,146)(44,147,109,215,190,125,87)(45,88,126,191,216,110,148)(46,149,111,209,192,127,81)(47,82,128,185,210,112,150)(48,151,105,211,186,121,83)(49,207,135,104,161,66,140)(50,141,67,162,97,136,208)(51,201,129,98,163,68,142)(52,143,69,164,99,130,202)(53,203,131,100,165,70,144)(54,137,71,166,101,132,204)(55,205,133,102,167,72,138)(56,139,65,168,103,134,206), (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), (1,156,43,161)(2,155,44,168)(3,154,45,167)(4,153,46,166)(5,160,47,165)(6,159,48,164)(7,158,41,163)(8,157,42,162)(9,112,131,35)(10,111,132,34)(11,110,133,33)(12,109,134,40)(13,108,135,39)(14,107,136,38)(15,106,129,37)(16,105,130,36)(17,81,71,179)(18,88,72,178)(19,87,65,177)(20,86,66,184)(21,85,67,183)(22,84,68,182)(23,83,69,181)(24,82,70,180)(25,171,123,141)(26,170,124,140)(27,169,125,139)(28,176,126,138)(29,175,127,137)(30,174,128,144)(31,173,121,143)(32,172,122,142)(49,77,61,189)(50,76,62,188)(51,75,63,187)(52,74,64,186)(53,73,57,185)(54,80,58,192)(55,79,59,191)(56,78,60,190)(89,210,203,119)(90,209,204,118)(91,216,205,117)(92,215,206,116)(93,214,207,115)(94,213,208,114)(95,212,201,113)(96,211,202,120)(97,224,200,145)(98,223,193,152)(99,222,194,151)(100,221,195,150)(101,220,196,149)(102,219,197,148)(103,218,198,147)(104,217,199,146), (2,6)(4,8)(9,131)(10,136)(11,133)(12,130)(13,135)(14,132)(15,129)(16,134)(17,67)(18,72)(19,69)(20,66)(21,71)(22,68)(23,65)(24,70)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(49,61)(50,58)(51,63)(52,60)(53,57)(54,62)(55,59)(56,64)(74,78)(76,80)(81,85)(83,87)(89,203)(90,208)(91,205)(92,202)(93,207)(94,204)(95,201)(96,206)(97,196)(98,193)(99,198)(100,195)(101,200)(102,197)(103,194)(104,199)(105,109)(107,111)(114,118)(116,120)(121,125)(123,127)(137,171)(138,176)(139,173)(140,170)(141,175)(142,172)(143,169)(144,174)(145,149)(147,151)(153,162)(154,167)(155,164)(156,161)(157,166)(158,163)(159,168)(160,165)(177,181)(179,183)(186,190)(188,192)(209,213)(211,215)(218,222)(220,224) );
G=PermutationGroup([[(1,184,26,77,115,39,217),(2,218,40,116,78,27,177),(3,178,28,79,117,33,219),(4,220,34,118,80,29,179),(5,180,30,73,119,35,221),(6,222,36,120,74,31,181),(7,182,32,75,113,37,223),(8,224,38,114,76,25,183),(9,195,160,24,174,57,89),(10,90,58,175,17,153,196),(11,197,154,18,176,59,91),(12,92,60,169,19,155,198),(13,199,156,20,170,61,93),(14,94,62,171,21,157,200),(15,193,158,22,172,63,95),(16,96,64,173,23,159,194),(41,84,122,187,212,106,152),(42,145,107,213,188,123,85),(43,86,124,189,214,108,146),(44,147,109,215,190,125,87),(45,88,126,191,216,110,148),(46,149,111,209,192,127,81),(47,82,128,185,210,112,150),(48,151,105,211,186,121,83),(49,207,135,104,161,66,140),(50,141,67,162,97,136,208),(51,201,129,98,163,68,142),(52,143,69,164,99,130,202),(53,203,131,100,165,70,144),(54,137,71,166,101,132,204),(55,205,133,102,167,72,138),(56,139,65,168,103,134,206)], [(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)], [(1,156,43,161),(2,155,44,168),(3,154,45,167),(4,153,46,166),(5,160,47,165),(6,159,48,164),(7,158,41,163),(8,157,42,162),(9,112,131,35),(10,111,132,34),(11,110,133,33),(12,109,134,40),(13,108,135,39),(14,107,136,38),(15,106,129,37),(16,105,130,36),(17,81,71,179),(18,88,72,178),(19,87,65,177),(20,86,66,184),(21,85,67,183),(22,84,68,182),(23,83,69,181),(24,82,70,180),(25,171,123,141),(26,170,124,140),(27,169,125,139),(28,176,126,138),(29,175,127,137),(30,174,128,144),(31,173,121,143),(32,172,122,142),(49,77,61,189),(50,76,62,188),(51,75,63,187),(52,74,64,186),(53,73,57,185),(54,80,58,192),(55,79,59,191),(56,78,60,190),(89,210,203,119),(90,209,204,118),(91,216,205,117),(92,215,206,116),(93,214,207,115),(94,213,208,114),(95,212,201,113),(96,211,202,120),(97,224,200,145),(98,223,193,152),(99,222,194,151),(100,221,195,150),(101,220,196,149),(102,219,197,148),(103,218,198,147),(104,217,199,146)], [(2,6),(4,8),(9,131),(10,136),(11,133),(12,130),(13,135),(14,132),(15,129),(16,134),(17,67),(18,72),(19,69),(20,66),(21,71),(22,68),(23,65),(24,70),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(49,61),(50,58),(51,63),(52,60),(53,57),(54,62),(55,59),(56,64),(74,78),(76,80),(81,85),(83,87),(89,203),(90,208),(91,205),(92,202),(93,207),(94,204),(95,201),(96,206),(97,196),(98,193),(99,198),(100,195),(101,200),(102,197),(103,194),(104,199),(105,109),(107,111),(114,118),(116,120),(121,125),(123,127),(137,171),(138,176),(139,173),(140,170),(141,175),(142,172),(143,169),(144,174),(145,149),(147,151),(153,162),(154,167),(155,164),(156,161),(157,166),(158,163),(159,168),(160,165),(177,181),(179,183),(186,190),(188,192),(209,213),(211,215),(218,222),(220,224)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 4 | 8 | 56 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8⋊C22 | C8.C22 | D4×D7 | D4⋊2D7 | D4.D14 | D4.9D14 |
| kernel | C7⋊C8⋊5D4 | C28.Q8 | C14.Q16 | D4⋊Dic7 | C2×C4.Dic7 | C28.48D4 | C2×D4.D7 | C7×C4⋊D4 | C7⋊C8 | C2×C28 | C22×C14 | C4⋊D4 | C28 | C4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C23 | C14 | C14 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 6 | 6 | 1 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of C7⋊C8⋊5D4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 112 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 75 | 32 |
| 0 | 0 | 0 | 0 | 29 | 38 |
| 0 | 0 | 29 | 40 | 0 | 0 |
| 0 | 0 | 75 | 84 | 0 | 0 |
| 111 | 106 | 0 | 0 | 0 | 0 |
| 33 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 49 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,1,0,0,0,0,112,0,0,0,0,0,0,0,9,1,0,0,0,0,112,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,29,75,0,0,0,0,40,84,0,0,75,29,0,0,0,0,32,38,0,0],[111,33,0,0,0,0,106,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,112,0,0,0,0,0,0,112,0,0],[112,49,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;
C7⋊C8⋊5D4 in GAP, Magma, Sage, TeX
C_7\rtimes C_8\rtimes_5D_4
% in TeX
G:=Group("C7:C8:5D4"); // GroupNames label
G:=SmallGroup(448,576);
// by ID
G=gap.SmallGroup(448,576);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,555,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations