direct product, metabelian, supersoluble, monomial
Aliases: C4×C32⋊7D4, C62.249C23, (C3×C12)⋊25D4, C32⋊23(C4×D4), C62⋊19(C2×C4), C12⋊13(C3⋊D4), (C22×C12)⋊13S3, (C2×C12).360D6, C62⋊5C4⋊22C2, (C22×C6).158D6, C6.107(C4○D12), C6.11D12⋊29C2, (C6×C12).360C22, C6.Dic6⋊29C2, C2.5(C12.59D6), (C2×C62).110C22, (C2×C6×C12)⋊15C2, C3⋊7(C4×C3⋊D4), C6.80(S3×C2×C4), (C2×C6)⋊14(C4×S3), C22⋊3(C4×C3⋊S3), C3⋊Dic3⋊9(C2×C4), (C22×C4)⋊4(C3⋊S3), (C4×C3⋊Dic3)⋊27C2, (C3×C6).275(C2×D4), C6.116(C2×C3⋊D4), C23.27(C2×C3⋊S3), C2.3(C2×C32⋊7D4), (C3×C6).122(C4○D4), (C2×C6).266(C22×S3), (C3×C6).111(C22×C4), (C2×C32⋊7D4).14C2, C22.24(C22×C3⋊S3), (C22×C3⋊S3).91C22, (C2×C3⋊Dic3).164C22, (C2×C4×C3⋊S3)⋊23C2, C2.20(C2×C4×C3⋊S3), (C2×C3⋊S3)⋊12(C2×C4), (C2×C4).103(C2×C3⋊S3), SmallGroup(288,785)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C32⋊7D4
G = < a,b,c,d,e | a4=b3=c3=d4=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: 932 in 282 conjugacy classes, 97 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4×D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, C6×C12, C22×C3⋊S3, C2×C62, C4×C3⋊D4, C4×C3⋊Dic3, C6.Dic6, C6.11D12, C62⋊5C4, C2×C4×C3⋊S3, C2×C32⋊7D4, C2×C6×C12, C4×C32⋊7D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C3⋊S3, C4×S3, C3⋊D4, C22×S3, C4×D4, C2×C3⋊S3, S3×C2×C4, C4○D12, C2×C3⋊D4, C4×C3⋊S3, C32⋊7D4, C22×C3⋊S3, C4×C3⋊D4, C2×C4×C3⋊S3, C12.59D6, C2×C32⋊7D4, C4×C32⋊7D4
►On 144 points
Generators in S144
(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)
(1 23 15)(2 24 16)(3 21 13)(4 22 14)(5 111 57)(6 112 58)(7 109 59)(8 110 60)(9 29 77)(10 30 78)(11 31 79)(12 32 80)(17 82 115)(18 83 116)(19 84 113)(20 81 114)(25 69 43)(26 70 44)(27 71 41)(28 72 42)(33 49 47)(34 50 48)(35 51 45)(36 52 46)(37 128 122)(38 125 123)(39 126 124)(40 127 121)(53 118 90)(54 119 91)(55 120 92)(56 117 89)(61 75 67)(62 76 68)(63 73 65)(64 74 66)(85 143 93)(86 144 94)(87 141 95)(88 142 96)(97 131 103)(98 132 104)(99 129 101)(100 130 102)(105 139 133)(106 140 134)(107 137 135)(108 138 136)
(1 41 35)(2 42 36)(3 43 33)(4 44 34)(5 67 79)(6 68 80)(7 65 77)(8 66 78)(9 109 63)(10 110 64)(11 111 61)(12 112 62)(13 69 47)(14 70 48)(15 71 45)(16 72 46)(17 124 118)(18 121 119)(19 122 120)(20 123 117)(21 25 49)(22 26 50)(23 27 51)(24 28 52)(29 59 73)(30 60 74)(31 57 75)(32 58 76)(37 92 84)(38 89 81)(39 90 82)(40 91 83)(53 115 126)(54 116 127)(55 113 128)(56 114 125)(85 137 129)(86 138 130)(87 139 131)(88 140 132)(93 107 99)(94 108 100)(95 105 97)(96 106 98)(101 143 135)(102 144 136)(103 141 133)(104 142 134)
(1 99 73 40)(2 100 74 37)(3 97 75 38)(4 98 76 39)(5 117 49 141)(6 118 50 142)(7 119 51 143)(8 120 52 144)(9 116 71 137)(10 113 72 138)(11 114 69 139)(12 115 70 140)(13 131 61 125)(14 132 62 126)(15 129 63 127)(16 130 64 128)(17 26 134 80)(18 27 135 77)(19 28 136 78)(20 25 133 79)(21 103 67 123)(22 104 68 124)(23 101 65 121)(24 102 66 122)(29 83 41 107)(30 84 42 108)(31 81 43 105)(32 82 44 106)(33 95 57 89)(34 96 58 90)(35 93 59 91)(36 94 60 92)(45 85 109 54)(46 86 110 55)(47 87 111 56)(48 88 112 53)
(1 3)(2 4)(5 9)(6 10)(7 11)(8 12)(13 23)(14 24)(15 21)(16 22)(17 86)(18 87)(19 88)(20 85)(25 45)(26 46)(27 47)(28 48)(29 57)(30 58)(31 59)(32 60)(33 41)(34 42)(35 43)(36 44)(37 98)(38 99)(39 100)(40 97)(49 71)(50 72)(51 69)(52 70)(53 136)(54 133)(55 134)(56 135)(61 65)(62 66)(63 67)(64 68)(73 75)(74 76)(77 111)(78 112)(79 109)(80 110)(81 93)(82 94)(83 95)(84 96)(89 107)(90 108)(91 105)(92 106)(101 125)(102 126)(103 127)(104 128)(113 142)(114 143)(115 144)(116 141)(117 137)(118 138)(119 139)(120 140)(121 131)(122 132)(123 129)(124 130)
G:=sub<Sym(144)| (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), (1,23,15)(2,24,16)(3,21,13)(4,22,14)(5,111,57)(6,112,58)(7,109,59)(8,110,60)(9,29,77)(10,30,78)(11,31,79)(12,32,80)(17,82,115)(18,83,116)(19,84,113)(20,81,114)(25,69,43)(26,70,44)(27,71,41)(28,72,42)(33,49,47)(34,50,48)(35,51,45)(36,52,46)(37,128,122)(38,125,123)(39,126,124)(40,127,121)(53,118,90)(54,119,91)(55,120,92)(56,117,89)(61,75,67)(62,76,68)(63,73,65)(64,74,66)(85,143,93)(86,144,94)(87,141,95)(88,142,96)(97,131,103)(98,132,104)(99,129,101)(100,130,102)(105,139,133)(106,140,134)(107,137,135)(108,138,136), (1,41,35)(2,42,36)(3,43,33)(4,44,34)(5,67,79)(6,68,80)(7,65,77)(8,66,78)(9,109,63)(10,110,64)(11,111,61)(12,112,62)(13,69,47)(14,70,48)(15,71,45)(16,72,46)(17,124,118)(18,121,119)(19,122,120)(20,123,117)(21,25,49)(22,26,50)(23,27,51)(24,28,52)(29,59,73)(30,60,74)(31,57,75)(32,58,76)(37,92,84)(38,89,81)(39,90,82)(40,91,83)(53,115,126)(54,116,127)(55,113,128)(56,114,125)(85,137,129)(86,138,130)(87,139,131)(88,140,132)(93,107,99)(94,108,100)(95,105,97)(96,106,98)(101,143,135)(102,144,136)(103,141,133)(104,142,134), (1,99,73,40)(2,100,74,37)(3,97,75,38)(4,98,76,39)(5,117,49,141)(6,118,50,142)(7,119,51,143)(8,120,52,144)(9,116,71,137)(10,113,72,138)(11,114,69,139)(12,115,70,140)(13,131,61,125)(14,132,62,126)(15,129,63,127)(16,130,64,128)(17,26,134,80)(18,27,135,77)(19,28,136,78)(20,25,133,79)(21,103,67,123)(22,104,68,124)(23,101,65,121)(24,102,66,122)(29,83,41,107)(30,84,42,108)(31,81,43,105)(32,82,44,106)(33,95,57,89)(34,96,58,90)(35,93,59,91)(36,94,60,92)(45,85,109,54)(46,86,110,55)(47,87,111,56)(48,88,112,53), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,23)(14,24)(15,21)(16,22)(17,86)(18,87)(19,88)(20,85)(25,45)(26,46)(27,47)(28,48)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,98)(38,99)(39,100)(40,97)(49,71)(50,72)(51,69)(52,70)(53,136)(54,133)(55,134)(56,135)(61,65)(62,66)(63,67)(64,68)(73,75)(74,76)(77,111)(78,112)(79,109)(80,110)(81,93)(82,94)(83,95)(84,96)(89,107)(90,108)(91,105)(92,106)(101,125)(102,126)(103,127)(104,128)(113,142)(114,143)(115,144)(116,141)(117,137)(118,138)(119,139)(120,140)(121,131)(122,132)(123,129)(124,130)>;
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), (1,23,15)(2,24,16)(3,21,13)(4,22,14)(5,111,57)(6,112,58)(7,109,59)(8,110,60)(9,29,77)(10,30,78)(11,31,79)(12,32,80)(17,82,115)(18,83,116)(19,84,113)(20,81,114)(25,69,43)(26,70,44)(27,71,41)(28,72,42)(33,49,47)(34,50,48)(35,51,45)(36,52,46)(37,128,122)(38,125,123)(39,126,124)(40,127,121)(53,118,90)(54,119,91)(55,120,92)(56,117,89)(61,75,67)(62,76,68)(63,73,65)(64,74,66)(85,143,93)(86,144,94)(87,141,95)(88,142,96)(97,131,103)(98,132,104)(99,129,101)(100,130,102)(105,139,133)(106,140,134)(107,137,135)(108,138,136), (1,41,35)(2,42,36)(3,43,33)(4,44,34)(5,67,79)(6,68,80)(7,65,77)(8,66,78)(9,109,63)(10,110,64)(11,111,61)(12,112,62)(13,69,47)(14,70,48)(15,71,45)(16,72,46)(17,124,118)(18,121,119)(19,122,120)(20,123,117)(21,25,49)(22,26,50)(23,27,51)(24,28,52)(29,59,73)(30,60,74)(31,57,75)(32,58,76)(37,92,84)(38,89,81)(39,90,82)(40,91,83)(53,115,126)(54,116,127)(55,113,128)(56,114,125)(85,137,129)(86,138,130)(87,139,131)(88,140,132)(93,107,99)(94,108,100)(95,105,97)(96,106,98)(101,143,135)(102,144,136)(103,141,133)(104,142,134), (1,99,73,40)(2,100,74,37)(3,97,75,38)(4,98,76,39)(5,117,49,141)(6,118,50,142)(7,119,51,143)(8,120,52,144)(9,116,71,137)(10,113,72,138)(11,114,69,139)(12,115,70,140)(13,131,61,125)(14,132,62,126)(15,129,63,127)(16,130,64,128)(17,26,134,80)(18,27,135,77)(19,28,136,78)(20,25,133,79)(21,103,67,123)(22,104,68,124)(23,101,65,121)(24,102,66,122)(29,83,41,107)(30,84,42,108)(31,81,43,105)(32,82,44,106)(33,95,57,89)(34,96,58,90)(35,93,59,91)(36,94,60,92)(45,85,109,54)(46,86,110,55)(47,87,111,56)(48,88,112,53), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,23)(14,24)(15,21)(16,22)(17,86)(18,87)(19,88)(20,85)(25,45)(26,46)(27,47)(28,48)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,98)(38,99)(39,100)(40,97)(49,71)(50,72)(51,69)(52,70)(53,136)(54,133)(55,134)(56,135)(61,65)(62,66)(63,67)(64,68)(73,75)(74,76)(77,111)(78,112)(79,109)(80,110)(81,93)(82,94)(83,95)(84,96)(89,107)(90,108)(91,105)(92,106)(101,125)(102,126)(103,127)(104,128)(113,142)(114,143)(115,144)(116,141)(117,137)(118,138)(119,139)(120,140)(121,131)(122,132)(123,129)(124,130) );
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)], [(1,23,15),(2,24,16),(3,21,13),(4,22,14),(5,111,57),(6,112,58),(7,109,59),(8,110,60),(9,29,77),(10,30,78),(11,31,79),(12,32,80),(17,82,115),(18,83,116),(19,84,113),(20,81,114),(25,69,43),(26,70,44),(27,71,41),(28,72,42),(33,49,47),(34,50,48),(35,51,45),(36,52,46),(37,128,122),(38,125,123),(39,126,124),(40,127,121),(53,118,90),(54,119,91),(55,120,92),(56,117,89),(61,75,67),(62,76,68),(63,73,65),(64,74,66),(85,143,93),(86,144,94),(87,141,95),(88,142,96),(97,131,103),(98,132,104),(99,129,101),(100,130,102),(105,139,133),(106,140,134),(107,137,135),(108,138,136)], [(1,41,35),(2,42,36),(3,43,33),(4,44,34),(5,67,79),(6,68,80),(7,65,77),(8,66,78),(9,109,63),(10,110,64),(11,111,61),(12,112,62),(13,69,47),(14,70,48),(15,71,45),(16,72,46),(17,124,118),(18,121,119),(19,122,120),(20,123,117),(21,25,49),(22,26,50),(23,27,51),(24,28,52),(29,59,73),(30,60,74),(31,57,75),(32,58,76),(37,92,84),(38,89,81),(39,90,82),(40,91,83),(53,115,126),(54,116,127),(55,113,128),(56,114,125),(85,137,129),(86,138,130),(87,139,131),(88,140,132),(93,107,99),(94,108,100),(95,105,97),(96,106,98),(101,143,135),(102,144,136),(103,141,133),(104,142,134)], [(1,99,73,40),(2,100,74,37),(3,97,75,38),(4,98,76,39),(5,117,49,141),(6,118,50,142),(7,119,51,143),(8,120,52,144),(9,116,71,137),(10,113,72,138),(11,114,69,139),(12,115,70,140),(13,131,61,125),(14,132,62,126),(15,129,63,127),(16,130,64,128),(17,26,134,80),(18,27,135,77),(19,28,136,78),(20,25,133,79),(21,103,67,123),(22,104,68,124),(23,101,65,121),(24,102,66,122),(29,83,41,107),(30,84,42,108),(31,81,43,105),(32,82,44,106),(33,95,57,89),(34,96,58,90),(35,93,59,91),(36,94,60,92),(45,85,109,54),(46,86,110,55),(47,87,111,56),(48,88,112,53)], [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12),(13,23),(14,24),(15,21),(16,22),(17,86),(18,87),(19,88),(20,85),(25,45),(26,46),(27,47),(28,48),(29,57),(30,58),(31,59),(32,60),(33,41),(34,42),(35,43),(36,44),(37,98),(38,99),(39,100),(40,97),(49,71),(50,72),(51,69),(52,70),(53,136),(54,133),(55,134),(56,135),(61,65),(62,66),(63,67),(64,68),(73,75),(74,76),(77,111),(78,112),(79,109),(80,110),(81,93),(82,94),(83,95),(84,96),(89,107),(90,108),(91,105),(92,106),(101,125),(102,126),(103,127),(104,128),(113,142),(114,143),(115,144),(116,141),(117,137),(118,138),(119,139),(120,140),(121,131),(122,132),(123,129),(124,130)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 6A | ··· | 6AB | 12A | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4×S3 | C4○D12 |
| kernel | C4×C32⋊7D4 | C4×C3⋊Dic3 | C6.Dic6 | C6.11D12 | C62⋊5C4 | C2×C4×C3⋊S3 | C2×C32⋊7D4 | C2×C6×C12 | C32⋊7D4 | C22×C12 | C3×C12 | C2×C12 | C22×C6 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 4 | 2 | 8 | 4 | 2 | 16 | 16 | 16 |
Matrix representation of C4×C32⋊7D4 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 12 | 0 |
| 2 | 4 | 0 | 0 |
| 2 | 11 | 0 | 0 |
| 0 | 0 | 2 | 9 |
| 0 | 0 | 11 | 11 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 1 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,1,0],[0,12,0,0,1,12,0,0,0,0,12,12,0,0,1,0],[2,2,0,0,4,11,0,0,0,0,2,11,0,0,9,11],[1,12,0,0,0,12,0,0,0,0,12,12,0,0,0,1] >;
C4×C32⋊7D4 in GAP, Magma, Sage, TeX
C_4\times C_3^2\rtimes_7D_4
% in TeX
G:=Group("C4xC3^2:7D4"); // GroupNames label
G:=SmallGroup(288,785);
// by ID
G=gap.SmallGroup(288,785);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^4=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