direct product, metabelian, supersoluble, monomial
Aliases: C2×C32⋊7D8, C62.130D4, (C3×C6)⋊7D8, (C6×D4)⋊1S3, C6⋊3(D4⋊S3), (C3×D4)⋊14D6, C32⋊14(C2×D8), (C3×C12).97D4, (C2×C12).149D6, C12.56(C3⋊D4), C12.97(C22×S3), C4.5(C32⋊7D4), C12⋊S3⋊18C22, (C6×C12).140C22, (C3×C12).101C23, C32⋊4C8⋊22C22, (D4×C32)⋊16C22, C22.21(C32⋊7D4), (D4×C3×C6)⋊5C2, D4⋊3(C2×C3⋊S3), C3⋊4(C2×D4⋊S3), (C2×D4)⋊1(C3⋊S3), (C2×C32⋊4C8)⋊9C2, (C3×C6).278(C2×D4), C6.119(C2×C3⋊D4), (C2×C12⋊S3)⋊14C2, C4.11(C22×C3⋊S3), C2.8(C2×C32⋊7D4), (C2×C6).98(C3⋊D4), (C2×C4).46(C2×C3⋊S3), SmallGroup(288,788)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32⋊7D8
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 980 in 228 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, D4, C23, C32, C12, D6, C2×C6, C2×C6, C2×C8, D8, C2×D4, C2×D4, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, D12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C2×D8, C3×C12, C2×C3⋊S3, C62, C62, C2×C3⋊C8, D4⋊S3, C2×D12, C6×D4, C32⋊4C8, C12⋊S3, C12⋊S3, C6×C12, D4×C32, D4×C32, C22×C3⋊S3, C2×C62, C2×D4⋊S3, C2×C32⋊4C8, C32⋊7D8, C2×C12⋊S3, D4×C3×C6, C2×C32⋊7D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C2×D8, C2×C3⋊S3, D4⋊S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, C2×D4⋊S3, C32⋊7D8, C2×C32⋊7D4, C2×C32⋊7D8
►On 144 points
Generators in S144
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 140)(10 141)(11 142)(12 143)(13 144)(14 137)(15 138)(16 139)(17 84)(18 85)(19 86)(20 87)(21 88)(22 81)(23 82)(24 83)(25 80)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 68)(34 69)(35 70)(36 71)(37 72)(38 65)(39 66)(40 67)(41 96)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 97)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 127)(114 128)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)
(1 9 55)(2 56 10)(3 11 49)(4 50 12)(5 13 51)(6 52 14)(7 15 53)(8 54 16)(17 127 44)(18 45 128)(19 121 46)(20 47 122)(21 123 48)(22 41 124)(23 125 42)(24 43 126)(25 68 106)(26 107 69)(27 70 108)(28 109 71)(29 72 110)(30 111 65)(31 66 112)(32 105 67)(33 130 80)(34 73 131)(35 132 74)(36 75 133)(37 134 76)(38 77 135)(39 136 78)(40 79 129)(57 101 137)(58 138 102)(59 103 139)(60 140 104)(61 97 141)(62 142 98)(63 99 143)(64 144 100)(81 96 118)(82 119 89)(83 90 120)(84 113 91)(85 92 114)(86 115 93)(87 94 116)(88 117 95)
(1 20 25)(2 26 21)(3 22 27)(4 28 23)(5 24 29)(6 30 17)(7 18 31)(8 32 19)(9 47 68)(10 69 48)(11 41 70)(12 71 42)(13 43 72)(14 65 44)(15 45 66)(16 67 46)(33 140 94)(34 95 141)(35 142 96)(36 89 143)(37 144 90)(38 91 137)(39 138 92)(40 93 139)(49 124 108)(50 109 125)(51 126 110)(52 111 127)(53 128 112)(54 105 121)(55 122 106)(56 107 123)(57 77 84)(58 85 78)(59 79 86)(60 87 80)(61 73 88)(62 81 74)(63 75 82)(64 83 76)(97 131 117)(98 118 132)(99 133 119)(100 120 134)(101 135 113)(102 114 136)(103 129 115)(104 116 130)
(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)
(2 8)(3 7)(4 6)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 56)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)(33 116)(34 115)(35 114)(36 113)(37 120)(38 119)(39 118)(40 117)(41 112)(42 111)(43 110)(44 109)(45 108)(46 107)(47 106)(48 105)(57 63)(58 62)(59 61)(65 125)(66 124)(67 123)(68 122)(69 121)(70 128)(71 127)(72 126)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 88)(80 87)(89 135)(90 134)(91 133)(92 132)(93 131)(94 130)(95 129)(96 136)(97 139)(98 138)(99 137)(100 144)(101 143)(102 142)(103 141)(104 140)
G:=sub<Sym(144)| (1,60)(2,61)(3,62)(4,63)(5,64)(6,57)(7,58)(8,59)(9,140)(10,141)(11,142)(12,143)(13,144)(14,137)(15,138)(16,139)(17,84)(18,85)(19,86)(20,87)(21,88)(22,81)(23,82)(24,83)(25,80)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,68)(34,69)(35,70)(36,71)(37,72)(38,65)(39,66)(40,67)(41,96)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,97)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,127)(114,128)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (1,9,55)(2,56,10)(3,11,49)(4,50,12)(5,13,51)(6,52,14)(7,15,53)(8,54,16)(17,127,44)(18,45,128)(19,121,46)(20,47,122)(21,123,48)(22,41,124)(23,125,42)(24,43,126)(25,68,106)(26,107,69)(27,70,108)(28,109,71)(29,72,110)(30,111,65)(31,66,112)(32,105,67)(33,130,80)(34,73,131)(35,132,74)(36,75,133)(37,134,76)(38,77,135)(39,136,78)(40,79,129)(57,101,137)(58,138,102)(59,103,139)(60,140,104)(61,97,141)(62,142,98)(63,99,143)(64,144,100)(81,96,118)(82,119,89)(83,90,120)(84,113,91)(85,92,114)(86,115,93)(87,94,116)(88,117,95), (1,20,25)(2,26,21)(3,22,27)(4,28,23)(5,24,29)(6,30,17)(7,18,31)(8,32,19)(9,47,68)(10,69,48)(11,41,70)(12,71,42)(13,43,72)(14,65,44)(15,45,66)(16,67,46)(33,140,94)(34,95,141)(35,142,96)(36,89,143)(37,144,90)(38,91,137)(39,138,92)(40,93,139)(49,124,108)(50,109,125)(51,126,110)(52,111,127)(53,128,112)(54,105,121)(55,122,106)(56,107,123)(57,77,84)(58,85,78)(59,79,86)(60,87,80)(61,73,88)(62,81,74)(63,75,82)(64,83,76)(97,131,117)(98,118,132)(99,133,119)(100,120,134)(101,135,113)(102,114,136)(103,129,115)(104,116,130), (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), (2,8)(3,7)(4,6)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,56)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,116)(34,115)(35,114)(36,113)(37,120)(38,119)(39,118)(40,117)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(57,63)(58,62)(59,61)(65,125)(66,124)(67,123)(68,122)(69,121)(70,128)(71,127)(72,126)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,88)(80,87)(89,135)(90,134)(91,133)(92,132)(93,131)(94,130)(95,129)(96,136)(97,139)(98,138)(99,137)(100,144)(101,143)(102,142)(103,141)(104,140)>;
G:=Group( (1,60)(2,61)(3,62)(4,63)(5,64)(6,57)(7,58)(8,59)(9,140)(10,141)(11,142)(12,143)(13,144)(14,137)(15,138)(16,139)(17,84)(18,85)(19,86)(20,87)(21,88)(22,81)(23,82)(24,83)(25,80)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,68)(34,69)(35,70)(36,71)(37,72)(38,65)(39,66)(40,67)(41,96)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,97)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,127)(114,128)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (1,9,55)(2,56,10)(3,11,49)(4,50,12)(5,13,51)(6,52,14)(7,15,53)(8,54,16)(17,127,44)(18,45,128)(19,121,46)(20,47,122)(21,123,48)(22,41,124)(23,125,42)(24,43,126)(25,68,106)(26,107,69)(27,70,108)(28,109,71)(29,72,110)(30,111,65)(31,66,112)(32,105,67)(33,130,80)(34,73,131)(35,132,74)(36,75,133)(37,134,76)(38,77,135)(39,136,78)(40,79,129)(57,101,137)(58,138,102)(59,103,139)(60,140,104)(61,97,141)(62,142,98)(63,99,143)(64,144,100)(81,96,118)(82,119,89)(83,90,120)(84,113,91)(85,92,114)(86,115,93)(87,94,116)(88,117,95), (1,20,25)(2,26,21)(3,22,27)(4,28,23)(5,24,29)(6,30,17)(7,18,31)(8,32,19)(9,47,68)(10,69,48)(11,41,70)(12,71,42)(13,43,72)(14,65,44)(15,45,66)(16,67,46)(33,140,94)(34,95,141)(35,142,96)(36,89,143)(37,144,90)(38,91,137)(39,138,92)(40,93,139)(49,124,108)(50,109,125)(51,126,110)(52,111,127)(53,128,112)(54,105,121)(55,122,106)(56,107,123)(57,77,84)(58,85,78)(59,79,86)(60,87,80)(61,73,88)(62,81,74)(63,75,82)(64,83,76)(97,131,117)(98,118,132)(99,133,119)(100,120,134)(101,135,113)(102,114,136)(103,129,115)(104,116,130), (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), (2,8)(3,7)(4,6)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,56)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,116)(34,115)(35,114)(36,113)(37,120)(38,119)(39,118)(40,117)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(57,63)(58,62)(59,61)(65,125)(66,124)(67,123)(68,122)(69,121)(70,128)(71,127)(72,126)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,88)(80,87)(89,135)(90,134)(91,133)(92,132)(93,131)(94,130)(95,129)(96,136)(97,139)(98,138)(99,137)(100,144)(101,143)(102,142)(103,141)(104,140) );
G=PermutationGroup([[(1,60),(2,61),(3,62),(4,63),(5,64),(6,57),(7,58),(8,59),(9,140),(10,141),(11,142),(12,143),(13,144),(14,137),(15,138),(16,139),(17,84),(18,85),(19,86),(20,87),(21,88),(22,81),(23,82),(24,83),(25,80),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,68),(34,69),(35,70),(36,71),(37,72),(38,65),(39,66),(40,67),(41,96),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,98),(50,99),(51,100),(52,101),(53,102),(54,103),(55,104),(56,97),(105,129),(106,130),(107,131),(108,132),(109,133),(110,134),(111,135),(112,136),(113,127),(114,128),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126)], [(1,9,55),(2,56,10),(3,11,49),(4,50,12),(5,13,51),(6,52,14),(7,15,53),(8,54,16),(17,127,44),(18,45,128),(19,121,46),(20,47,122),(21,123,48),(22,41,124),(23,125,42),(24,43,126),(25,68,106),(26,107,69),(27,70,108),(28,109,71),(29,72,110),(30,111,65),(31,66,112),(32,105,67),(33,130,80),(34,73,131),(35,132,74),(36,75,133),(37,134,76),(38,77,135),(39,136,78),(40,79,129),(57,101,137),(58,138,102),(59,103,139),(60,140,104),(61,97,141),(62,142,98),(63,99,143),(64,144,100),(81,96,118),(82,119,89),(83,90,120),(84,113,91),(85,92,114),(86,115,93),(87,94,116),(88,117,95)], [(1,20,25),(2,26,21),(3,22,27),(4,28,23),(5,24,29),(6,30,17),(7,18,31),(8,32,19),(9,47,68),(10,69,48),(11,41,70),(12,71,42),(13,43,72),(14,65,44),(15,45,66),(16,67,46),(33,140,94),(34,95,141),(35,142,96),(36,89,143),(37,144,90),(38,91,137),(39,138,92),(40,93,139),(49,124,108),(50,109,125),(51,126,110),(52,111,127),(53,128,112),(54,105,121),(55,122,106),(56,107,123),(57,77,84),(58,85,78),(59,79,86),(60,87,80),(61,73,88),(62,81,74),(63,75,82),(64,83,76),(97,131,117),(98,118,132),(99,133,119),(100,120,134),(101,135,113),(102,114,136),(103,129,115),(104,116,130)], [(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)], [(2,8),(3,7),(4,6),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,56),(17,28),(18,27),(19,26),(20,25),(21,32),(22,31),(23,30),(24,29),(33,116),(34,115),(35,114),(36,113),(37,120),(38,119),(39,118),(40,117),(41,112),(42,111),(43,110),(44,109),(45,108),(46,107),(47,106),(48,105),(57,63),(58,62),(59,61),(65,125),(66,124),(67,123),(68,122),(69,121),(70,128),(71,127),(72,126),(73,86),(74,85),(75,84),(76,83),(77,82),(78,81),(79,88),(80,87),(89,135),(90,134),(91,133),(92,132),(93,131),(94,130),(95,129),(96,136),(97,139),(98,138),(99,137),(100,144),(101,143),(102,142),(103,141),(104,140)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 6A | ··· | 6L | 6M | ··· | 6AB | 8A | 8B | 8C | 8D | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | C3⋊D4 | C3⋊D4 | D4⋊S3 |
| kernel | C2×C32⋊7D8 | C2×C32⋊4C8 | C32⋊7D8 | C2×C12⋊S3 | D4×C3×C6 | C6×D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 1 | 4 | 8 | 4 | 8 | 8 | 8 |
Matrix representation of C2×C32⋊7D8 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 72 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 43 | 0 | 0 |
| 0 | 0 | 30 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 16 |
| 0 | 0 | 0 | 0 | 41 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 2 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,13,30,0,0,0,0,43,60,0,0,0,0,0,0,41,41,0,0,0,0,16,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,2,0,0,0,0,0,72] >;
C2×C32⋊7D8 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes_7D_8
% in TeX
G:=Group("C2xC3^2:7D8"); // GroupNames label
G:=SmallGroup(288,788);
// by ID
G=gap.SmallGroup(288,788);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,675,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations