metabelian, supersoluble, monomial
Aliases: C72⋊7D4, C142⋊3C2, C14.16D14, (C2×C14)⋊2D7, C22⋊(C7⋊D7), C7⋊3(C7⋊D4), C7⋊Dic7⋊3C2, (C7×C14).15C22, (C2×C7⋊D7)⋊3C2, C2.5(C2×C7⋊D7), SmallGroup(392,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C72⋊7D4
G = < a,b,c,d | a7=b7=c4=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 580 in 80 conjugacy classes, 33 normal (9 characteristic)
C1, C2, C2, C4, C22, C22, C7, D4, D7, C14, C14, Dic7, D14, C2×C14, C72, C7⋊D4, C7⋊D7, C7×C14, C7×C14, C7⋊Dic7, C2×C7⋊D7, C142, C72⋊7D4
Quotients: C1, C2, C22, D4, D7, D14, C7⋊D4, C7⋊D7, C2×C7⋊D7, C72⋊7D4
►On 196 points
Generators in S196
(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)
(1 90 54 73 82 68 61)(2 91 55 74 83 69 62)(3 85 56 75 84 70 63)(4 86 50 76 78 64 57)(5 87 51 77 79 65 58)(6 88 52 71 80 66 59)(7 89 53 72 81 67 60)(8 196 49 15 29 36 22)(9 190 43 16 30 37 23)(10 191 44 17 31 38 24)(11 192 45 18 32 39 25)(12 193 46 19 33 40 26)(13 194 47 20 34 41 27)(14 195 48 21 35 42 28)(92 106 120 127 113 100 134)(93 107 121 128 114 101 135)(94 108 122 129 115 102 136)(95 109 123 130 116 103 137)(96 110 124 131 117 104 138)(97 111 125 132 118 105 139)(98 112 126 133 119 99 140)(141 154 176 162 155 169 183)(142 148 177 163 156 170 184)(143 149 178 164 157 171 185)(144 150 179 165 158 172 186)(145 151 180 166 159 173 187)(146 152 181 167 160 174 188)(147 153 182 168 161 175 189)
(1 17 117 177)(2 16 118 176)(3 15 119 182)(4 21 113 181)(5 20 114 180)(6 19 115 179)(7 18 116 178)(8 98 189 75)(9 97 183 74)(10 96 184 73)(11 95 185 72)(12 94 186 71)(13 93 187 77)(14 92 188 76)(22 112 175 84)(23 111 169 83)(24 110 170 82)(25 109 171 81)(26 108 172 80)(27 107 173 79)(28 106 174 78)(29 133 168 63)(30 132 162 62)(31 131 163 61)(32 130 164 60)(33 129 165 59)(34 128 166 58)(35 127 167 57)(36 126 161 70)(37 125 155 69)(38 124 156 68)(39 123 157 67)(40 122 158 66)(41 121 159 65)(42 120 160 64)(43 105 154 91)(44 104 148 90)(45 103 149 89)(46 102 150 88)(47 101 151 87)(48 100 152 86)(49 99 153 85)(50 195 134 146)(51 194 135 145)(52 193 136 144)(53 192 137 143)(54 191 138 142)(55 190 139 141)(56 196 140 147)
(1 177)(2 176)(3 182)(4 181)(5 180)(6 179)(7 178)(8 98)(9 97)(10 96)(11 95)(12 94)(13 93)(14 92)(15 119)(16 118)(17 117)(18 116)(19 115)(20 114)(21 113)(22 112)(23 111)(24 110)(25 109)(26 108)(27 107)(28 106)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(41 121)(42 120)(43 105)(44 104)(45 103)(46 102)(47 101)(48 100)(49 99)(50 146)(51 145)(52 144)(53 143)(54 142)(55 141)(56 147)(57 167)(58 166)(59 165)(60 164)(61 163)(62 162)(63 168)(64 160)(65 159)(66 158)(67 157)(68 156)(69 155)(70 161)(71 186)(72 185)(73 184)(74 183)(75 189)(76 188)(77 187)(78 174)(79 173)(80 172)(81 171)(82 170)(83 169)(84 175)(85 153)(86 152)(87 151)(88 150)(89 149)(90 148)(91 154)(134 195)(135 194)(136 193)(137 192)(138 191)(139 190)(140 196)
G:=sub<Sym(196)| (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), (1,90,54,73,82,68,61)(2,91,55,74,83,69,62)(3,85,56,75,84,70,63)(4,86,50,76,78,64,57)(5,87,51,77,79,65,58)(6,88,52,71,80,66,59)(7,89,53,72,81,67,60)(8,196,49,15,29,36,22)(9,190,43,16,30,37,23)(10,191,44,17,31,38,24)(11,192,45,18,32,39,25)(12,193,46,19,33,40,26)(13,194,47,20,34,41,27)(14,195,48,21,35,42,28)(92,106,120,127,113,100,134)(93,107,121,128,114,101,135)(94,108,122,129,115,102,136)(95,109,123,130,116,103,137)(96,110,124,131,117,104,138)(97,111,125,132,118,105,139)(98,112,126,133,119,99,140)(141,154,176,162,155,169,183)(142,148,177,163,156,170,184)(143,149,178,164,157,171,185)(144,150,179,165,158,172,186)(145,151,180,166,159,173,187)(146,152,181,167,160,174,188)(147,153,182,168,161,175,189), (1,17,117,177)(2,16,118,176)(3,15,119,182)(4,21,113,181)(5,20,114,180)(6,19,115,179)(7,18,116,178)(8,98,189,75)(9,97,183,74)(10,96,184,73)(11,95,185,72)(12,94,186,71)(13,93,187,77)(14,92,188,76)(22,112,175,84)(23,111,169,83)(24,110,170,82)(25,109,171,81)(26,108,172,80)(27,107,173,79)(28,106,174,78)(29,133,168,63)(30,132,162,62)(31,131,163,61)(32,130,164,60)(33,129,165,59)(34,128,166,58)(35,127,167,57)(36,126,161,70)(37,125,155,69)(38,124,156,68)(39,123,157,67)(40,122,158,66)(41,121,159,65)(42,120,160,64)(43,105,154,91)(44,104,148,90)(45,103,149,89)(46,102,150,88)(47,101,151,87)(48,100,152,86)(49,99,153,85)(50,195,134,146)(51,194,135,145)(52,193,136,144)(53,192,137,143)(54,191,138,142)(55,190,139,141)(56,196,140,147), (1,177)(2,176)(3,182)(4,181)(5,180)(6,179)(7,178)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,119)(16,118)(17,117)(18,116)(19,115)(20,114)(21,113)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,105)(44,104)(45,103)(46,102)(47,101)(48,100)(49,99)(50,146)(51,145)(52,144)(53,143)(54,142)(55,141)(56,147)(57,167)(58,166)(59,165)(60,164)(61,163)(62,162)(63,168)(64,160)(65,159)(66,158)(67,157)(68,156)(69,155)(70,161)(71,186)(72,185)(73,184)(74,183)(75,189)(76,188)(77,187)(78,174)(79,173)(80,172)(81,171)(82,170)(83,169)(84,175)(85,153)(86,152)(87,151)(88,150)(89,149)(90,148)(91,154)(134,195)(135,194)(136,193)(137,192)(138,191)(139,190)(140,196)>;
G:=Group( (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), (1,90,54,73,82,68,61)(2,91,55,74,83,69,62)(3,85,56,75,84,70,63)(4,86,50,76,78,64,57)(5,87,51,77,79,65,58)(6,88,52,71,80,66,59)(7,89,53,72,81,67,60)(8,196,49,15,29,36,22)(9,190,43,16,30,37,23)(10,191,44,17,31,38,24)(11,192,45,18,32,39,25)(12,193,46,19,33,40,26)(13,194,47,20,34,41,27)(14,195,48,21,35,42,28)(92,106,120,127,113,100,134)(93,107,121,128,114,101,135)(94,108,122,129,115,102,136)(95,109,123,130,116,103,137)(96,110,124,131,117,104,138)(97,111,125,132,118,105,139)(98,112,126,133,119,99,140)(141,154,176,162,155,169,183)(142,148,177,163,156,170,184)(143,149,178,164,157,171,185)(144,150,179,165,158,172,186)(145,151,180,166,159,173,187)(146,152,181,167,160,174,188)(147,153,182,168,161,175,189), (1,17,117,177)(2,16,118,176)(3,15,119,182)(4,21,113,181)(5,20,114,180)(6,19,115,179)(7,18,116,178)(8,98,189,75)(9,97,183,74)(10,96,184,73)(11,95,185,72)(12,94,186,71)(13,93,187,77)(14,92,188,76)(22,112,175,84)(23,111,169,83)(24,110,170,82)(25,109,171,81)(26,108,172,80)(27,107,173,79)(28,106,174,78)(29,133,168,63)(30,132,162,62)(31,131,163,61)(32,130,164,60)(33,129,165,59)(34,128,166,58)(35,127,167,57)(36,126,161,70)(37,125,155,69)(38,124,156,68)(39,123,157,67)(40,122,158,66)(41,121,159,65)(42,120,160,64)(43,105,154,91)(44,104,148,90)(45,103,149,89)(46,102,150,88)(47,101,151,87)(48,100,152,86)(49,99,153,85)(50,195,134,146)(51,194,135,145)(52,193,136,144)(53,192,137,143)(54,191,138,142)(55,190,139,141)(56,196,140,147), (1,177)(2,176)(3,182)(4,181)(5,180)(6,179)(7,178)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,119)(16,118)(17,117)(18,116)(19,115)(20,114)(21,113)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,121)(42,120)(43,105)(44,104)(45,103)(46,102)(47,101)(48,100)(49,99)(50,146)(51,145)(52,144)(53,143)(54,142)(55,141)(56,147)(57,167)(58,166)(59,165)(60,164)(61,163)(62,162)(63,168)(64,160)(65,159)(66,158)(67,157)(68,156)(69,155)(70,161)(71,186)(72,185)(73,184)(74,183)(75,189)(76,188)(77,187)(78,174)(79,173)(80,172)(81,171)(82,170)(83,169)(84,175)(85,153)(86,152)(87,151)(88,150)(89,149)(90,148)(91,154)(134,195)(135,194)(136,193)(137,192)(138,191)(139,190)(140,196) );
G=PermutationGroup([[(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)], [(1,90,54,73,82,68,61),(2,91,55,74,83,69,62),(3,85,56,75,84,70,63),(4,86,50,76,78,64,57),(5,87,51,77,79,65,58),(6,88,52,71,80,66,59),(7,89,53,72,81,67,60),(8,196,49,15,29,36,22),(9,190,43,16,30,37,23),(10,191,44,17,31,38,24),(11,192,45,18,32,39,25),(12,193,46,19,33,40,26),(13,194,47,20,34,41,27),(14,195,48,21,35,42,28),(92,106,120,127,113,100,134),(93,107,121,128,114,101,135),(94,108,122,129,115,102,136),(95,109,123,130,116,103,137),(96,110,124,131,117,104,138),(97,111,125,132,118,105,139),(98,112,126,133,119,99,140),(141,154,176,162,155,169,183),(142,148,177,163,156,170,184),(143,149,178,164,157,171,185),(144,150,179,165,158,172,186),(145,151,180,166,159,173,187),(146,152,181,167,160,174,188),(147,153,182,168,161,175,189)], [(1,17,117,177),(2,16,118,176),(3,15,119,182),(4,21,113,181),(5,20,114,180),(6,19,115,179),(7,18,116,178),(8,98,189,75),(9,97,183,74),(10,96,184,73),(11,95,185,72),(12,94,186,71),(13,93,187,77),(14,92,188,76),(22,112,175,84),(23,111,169,83),(24,110,170,82),(25,109,171,81),(26,108,172,80),(27,107,173,79),(28,106,174,78),(29,133,168,63),(30,132,162,62),(31,131,163,61),(32,130,164,60),(33,129,165,59),(34,128,166,58),(35,127,167,57),(36,126,161,70),(37,125,155,69),(38,124,156,68),(39,123,157,67),(40,122,158,66),(41,121,159,65),(42,120,160,64),(43,105,154,91),(44,104,148,90),(45,103,149,89),(46,102,150,88),(47,101,151,87),(48,100,152,86),(49,99,153,85),(50,195,134,146),(51,194,135,145),(52,193,136,144),(53,192,137,143),(54,191,138,142),(55,190,139,141),(56,196,140,147)], [(1,177),(2,176),(3,182),(4,181),(5,180),(6,179),(7,178),(8,98),(9,97),(10,96),(11,95),(12,94),(13,93),(14,92),(15,119),(16,118),(17,117),(18,116),(19,115),(20,114),(21,113),(22,112),(23,111),(24,110),(25,109),(26,108),(27,107),(28,106),(29,133),(30,132),(31,131),(32,130),(33,129),(34,128),(35,127),(36,126),(37,125),(38,124),(39,123),(40,122),(41,121),(42,120),(43,105),(44,104),(45,103),(46,102),(47,101),(48,100),(49,99),(50,146),(51,145),(52,144),(53,143),(54,142),(55,141),(56,147),(57,167),(58,166),(59,165),(60,164),(61,163),(62,162),(63,168),(64,160),(65,159),(66,158),(67,157),(68,156),(69,155),(70,161),(71,186),(72,185),(73,184),(74,183),(75,189),(76,188),(77,187),(78,174),(79,173),(80,172),(81,171),(82,170),(83,169),(84,175),(85,153),(86,152),(87,151),(88,150),(89,149),(90,148),(91,154),(134,195),(135,194),(136,193),(137,192),(138,191),(139,190),(140,196)]])
101 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 7A | ··· | 7X | 14A | ··· | 14BT |
| order | 1 | 2 | 2 | 2 | 4 | 7 | ··· | 7 | 14 | ··· | 14 |
| size | 1 | 1 | 2 | 98 | 98 | 2 | ··· | 2 | 2 | ··· | 2 |
101 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D7 | D14 | C7⋊D4 |
| kernel | C72⋊7D4 | C7⋊Dic7 | C2×C7⋊D7 | C142 | C72 | C2×C14 | C14 | C7 |
| # reps | 1 | 1 | 1 | 1 | 1 | 24 | 24 | 48 |
Matrix representation of C72⋊7D4 ►in GL4(𝔽29) generated by
| 16 | 26 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 8 | 7 |
| 24 | 8 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 23 | 23 |
| 2 | 20 | 0 | 0 |
| 7 | 27 | 0 | 0 |
| 0 | 0 | 22 | 23 |
| 0 | 0 | 18 | 7 |
| 27 | 17 | 0 | 0 |
| 22 | 2 | 0 | 0 |
| 0 | 0 | 22 | 23 |
| 0 | 0 | 8 | 7 |
G:=sub<GL(4,GF(29))| [16,0,0,0,26,20,0,0,0,0,25,8,0,0,0,7],[24,0,0,0,8,23,0,0,0,0,24,23,0,0,0,23],[2,7,0,0,20,27,0,0,0,0,22,18,0,0,23,7],[27,22,0,0,17,2,0,0,0,0,22,8,0,0,23,7] >;
C72⋊7D4 in GAP, Magma, Sage, TeX
C_7^2\rtimes_7D_4
% in TeX
G:=Group("C7^2:7D4"); // GroupNames label
G:=SmallGroup(392,32);
// by ID
G=gap.SmallGroup(392,32);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,61,963,8404]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^7=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations