direct product, metabelian, supersoluble, monomial, A-group
Aliases: C42×C3⋊S3, C122⋊15C2, C62.215C23, C12⋊10(C4×S3), (C4×C12)⋊10S3, C3⋊2(S3×C42), C32⋊6(C2×C42), (C2×C12).423D6, (C6×C12).354C22, C6.62(S3×C2×C4), (C3×C12)⋊20(C2×C4), C3⋊Dic3⋊18(C2×C4), (C4×C3⋊Dic3)⋊28C2, (C3×C6).93(C22×C4), C22.9(C22×C3⋊S3), (C2×C6).232(C22×S3), (C22×C3⋊S3).111C22, (C2×C3⋊Dic3).184C22, C2.1(C2×C4×C3⋊S3), (C2×C4×C3⋊S3).31C2, (C2×C4).96(C2×C3⋊S3), (C2×C3⋊S3).50(C2×C4), SmallGroup(288,728)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C42×C3⋊S3 |
Generators and relations for C42×C3⋊S3
G = < a,b,c,d,e | a4=b4=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 948 in 324 conjugacy classes, 129 normal (8 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C42, C42, C22×C4, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×C42, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C4×Dic3, C4×C12, S3×C2×C4, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C42, C4×C3⋊Dic3, C122, C2×C4×C3⋊S3, C42×C3⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, C22×C4, C3⋊S3, C4×S3, C22×S3, C2×C42, C2×C3⋊S3, S3×C2×C4, C4×C3⋊S3, C22×C3⋊S3, S3×C42, C2×C4×C3⋊S3, C42×C3⋊S3
►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 76 126 26)(2 73 127 27)(3 74 128 28)(4 75 125 25)(5 80 83 132)(6 77 84 129)(7 78 81 130)(8 79 82 131)(9 58 107 18)(10 59 108 19)(11 60 105 20)(12 57 106 17)(13 71 123 100)(14 72 124 97)(15 69 121 98)(16 70 122 99)(21 61 113 109)(22 62 114 110)(23 63 115 111)(24 64 116 112)(29 139 42 38)(30 140 43 39)(31 137 44 40)(32 138 41 37)(33 85 143 54)(34 86 144 55)(35 87 141 56)(36 88 142 53)(45 96 67 119)(46 93 68 120)(47 94 65 117)(48 95 66 118)(49 103 91 136)(50 104 92 133)(51 101 89 134)(52 102 90 135)
(1 114 15)(2 115 16)(3 116 13)(4 113 14)(5 139 136)(6 140 133)(7 137 134)(8 138 135)(9 47 141)(10 48 142)(11 45 143)(12 46 144)(17 120 86)(18 117 87)(19 118 88)(20 119 85)(21 124 125)(22 121 126)(23 122 127)(24 123 128)(25 61 97)(26 62 98)(27 63 99)(28 64 100)(29 91 132)(30 92 129)(31 89 130)(32 90 131)(33 105 67)(34 106 68)(35 107 65)(36 108 66)(37 102 82)(38 103 83)(39 104 84)(40 101 81)(41 52 79)(42 49 80)(43 50 77)(44 51 78)(53 59 95)(54 60 96)(55 57 93)(56 58 94)(69 76 110)(70 73 111)(71 74 112)(72 75 109)
(1 117 51)(2 118 52)(3 119 49)(4 120 50)(5 64 33)(6 61 34)(7 62 35)(8 63 36)(9 40 69)(10 37 70)(11 38 71)(12 39 72)(13 20 42)(14 17 43)(15 18 44)(16 19 41)(21 55 129)(22 56 130)(23 53 131)(24 54 132)(25 68 133)(26 65 134)(27 66 135)(28 67 136)(29 123 60)(30 124 57)(31 121 58)(32 122 59)(45 103 74)(46 104 75)(47 101 76)(48 102 73)(77 113 86)(78 114 87)(79 115 88)(80 116 85)(81 110 141)(82 111 142)(83 112 143)(84 109 144)(89 126 94)(90 127 95)(91 128 96)(92 125 93)(97 106 140)(98 107 137)(99 108 138)(100 105 139)
(1 126)(2 127)(3 128)(4 125)(5 11)(6 12)(7 9)(8 10)(13 24)(14 21)(15 22)(16 23)(17 129)(18 130)(19 131)(20 132)(25 75)(26 76)(27 73)(28 74)(29 85)(30 86)(31 87)(32 88)(33 38)(34 39)(35 40)(36 37)(41 53)(42 54)(43 55)(44 56)(45 136)(46 133)(47 134)(48 135)(49 96)(50 93)(51 94)(52 95)(57 77)(58 78)(59 79)(60 80)(61 72)(62 69)(63 70)(64 71)(65 101)(66 102)(67 103)(68 104)(81 107)(82 108)(83 105)(84 106)(89 117)(90 118)(91 119)(92 120)(97 109)(98 110)(99 111)(100 112)(113 124)(114 121)(115 122)(116 123)(137 141)(138 142)(139 143)(140 144)
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), (1,76,126,26)(2,73,127,27)(3,74,128,28)(4,75,125,25)(5,80,83,132)(6,77,84,129)(7,78,81,130)(8,79,82,131)(9,58,107,18)(10,59,108,19)(11,60,105,20)(12,57,106,17)(13,71,123,100)(14,72,124,97)(15,69,121,98)(16,70,122,99)(21,61,113,109)(22,62,114,110)(23,63,115,111)(24,64,116,112)(29,139,42,38)(30,140,43,39)(31,137,44,40)(32,138,41,37)(33,85,143,54)(34,86,144,55)(35,87,141,56)(36,88,142,53)(45,96,67,119)(46,93,68,120)(47,94,65,117)(48,95,66,118)(49,103,91,136)(50,104,92,133)(51,101,89,134)(52,102,90,135), (1,114,15)(2,115,16)(3,116,13)(4,113,14)(5,139,136)(6,140,133)(7,137,134)(8,138,135)(9,47,141)(10,48,142)(11,45,143)(12,46,144)(17,120,86)(18,117,87)(19,118,88)(20,119,85)(21,124,125)(22,121,126)(23,122,127)(24,123,128)(25,61,97)(26,62,98)(27,63,99)(28,64,100)(29,91,132)(30,92,129)(31,89,130)(32,90,131)(33,105,67)(34,106,68)(35,107,65)(36,108,66)(37,102,82)(38,103,83)(39,104,84)(40,101,81)(41,52,79)(42,49,80)(43,50,77)(44,51,78)(53,59,95)(54,60,96)(55,57,93)(56,58,94)(69,76,110)(70,73,111)(71,74,112)(72,75,109), (1,117,51)(2,118,52)(3,119,49)(4,120,50)(5,64,33)(6,61,34)(7,62,35)(8,63,36)(9,40,69)(10,37,70)(11,38,71)(12,39,72)(13,20,42)(14,17,43)(15,18,44)(16,19,41)(21,55,129)(22,56,130)(23,53,131)(24,54,132)(25,68,133)(26,65,134)(27,66,135)(28,67,136)(29,123,60)(30,124,57)(31,121,58)(32,122,59)(45,103,74)(46,104,75)(47,101,76)(48,102,73)(77,113,86)(78,114,87)(79,115,88)(80,116,85)(81,110,141)(82,111,142)(83,112,143)(84,109,144)(89,126,94)(90,127,95)(91,128,96)(92,125,93)(97,106,140)(98,107,137)(99,108,138)(100,105,139), (1,126)(2,127)(3,128)(4,125)(5,11)(6,12)(7,9)(8,10)(13,24)(14,21)(15,22)(16,23)(17,129)(18,130)(19,131)(20,132)(25,75)(26,76)(27,73)(28,74)(29,85)(30,86)(31,87)(32,88)(33,38)(34,39)(35,40)(36,37)(41,53)(42,54)(43,55)(44,56)(45,136)(46,133)(47,134)(48,135)(49,96)(50,93)(51,94)(52,95)(57,77)(58,78)(59,79)(60,80)(61,72)(62,69)(63,70)(64,71)(65,101)(66,102)(67,103)(68,104)(81,107)(82,108)(83,105)(84,106)(89,117)(90,118)(91,119)(92,120)(97,109)(98,110)(99,111)(100,112)(113,124)(114,121)(115,122)(116,123)(137,141)(138,142)(139,143)(140,144)>;
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), (1,76,126,26)(2,73,127,27)(3,74,128,28)(4,75,125,25)(5,80,83,132)(6,77,84,129)(7,78,81,130)(8,79,82,131)(9,58,107,18)(10,59,108,19)(11,60,105,20)(12,57,106,17)(13,71,123,100)(14,72,124,97)(15,69,121,98)(16,70,122,99)(21,61,113,109)(22,62,114,110)(23,63,115,111)(24,64,116,112)(29,139,42,38)(30,140,43,39)(31,137,44,40)(32,138,41,37)(33,85,143,54)(34,86,144,55)(35,87,141,56)(36,88,142,53)(45,96,67,119)(46,93,68,120)(47,94,65,117)(48,95,66,118)(49,103,91,136)(50,104,92,133)(51,101,89,134)(52,102,90,135), (1,114,15)(2,115,16)(3,116,13)(4,113,14)(5,139,136)(6,140,133)(7,137,134)(8,138,135)(9,47,141)(10,48,142)(11,45,143)(12,46,144)(17,120,86)(18,117,87)(19,118,88)(20,119,85)(21,124,125)(22,121,126)(23,122,127)(24,123,128)(25,61,97)(26,62,98)(27,63,99)(28,64,100)(29,91,132)(30,92,129)(31,89,130)(32,90,131)(33,105,67)(34,106,68)(35,107,65)(36,108,66)(37,102,82)(38,103,83)(39,104,84)(40,101,81)(41,52,79)(42,49,80)(43,50,77)(44,51,78)(53,59,95)(54,60,96)(55,57,93)(56,58,94)(69,76,110)(70,73,111)(71,74,112)(72,75,109), (1,117,51)(2,118,52)(3,119,49)(4,120,50)(5,64,33)(6,61,34)(7,62,35)(8,63,36)(9,40,69)(10,37,70)(11,38,71)(12,39,72)(13,20,42)(14,17,43)(15,18,44)(16,19,41)(21,55,129)(22,56,130)(23,53,131)(24,54,132)(25,68,133)(26,65,134)(27,66,135)(28,67,136)(29,123,60)(30,124,57)(31,121,58)(32,122,59)(45,103,74)(46,104,75)(47,101,76)(48,102,73)(77,113,86)(78,114,87)(79,115,88)(80,116,85)(81,110,141)(82,111,142)(83,112,143)(84,109,144)(89,126,94)(90,127,95)(91,128,96)(92,125,93)(97,106,140)(98,107,137)(99,108,138)(100,105,139), (1,126)(2,127)(3,128)(4,125)(5,11)(6,12)(7,9)(8,10)(13,24)(14,21)(15,22)(16,23)(17,129)(18,130)(19,131)(20,132)(25,75)(26,76)(27,73)(28,74)(29,85)(30,86)(31,87)(32,88)(33,38)(34,39)(35,40)(36,37)(41,53)(42,54)(43,55)(44,56)(45,136)(46,133)(47,134)(48,135)(49,96)(50,93)(51,94)(52,95)(57,77)(58,78)(59,79)(60,80)(61,72)(62,69)(63,70)(64,71)(65,101)(66,102)(67,103)(68,104)(81,107)(82,108)(83,105)(84,106)(89,117)(90,118)(91,119)(92,120)(97,109)(98,110)(99,111)(100,112)(113,124)(114,121)(115,122)(116,123)(137,141)(138,142)(139,143)(140,144) );
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)], [(1,76,126,26),(2,73,127,27),(3,74,128,28),(4,75,125,25),(5,80,83,132),(6,77,84,129),(7,78,81,130),(8,79,82,131),(9,58,107,18),(10,59,108,19),(11,60,105,20),(12,57,106,17),(13,71,123,100),(14,72,124,97),(15,69,121,98),(16,70,122,99),(21,61,113,109),(22,62,114,110),(23,63,115,111),(24,64,116,112),(29,139,42,38),(30,140,43,39),(31,137,44,40),(32,138,41,37),(33,85,143,54),(34,86,144,55),(35,87,141,56),(36,88,142,53),(45,96,67,119),(46,93,68,120),(47,94,65,117),(48,95,66,118),(49,103,91,136),(50,104,92,133),(51,101,89,134),(52,102,90,135)], [(1,114,15),(2,115,16),(3,116,13),(4,113,14),(5,139,136),(6,140,133),(7,137,134),(8,138,135),(9,47,141),(10,48,142),(11,45,143),(12,46,144),(17,120,86),(18,117,87),(19,118,88),(20,119,85),(21,124,125),(22,121,126),(23,122,127),(24,123,128),(25,61,97),(26,62,98),(27,63,99),(28,64,100),(29,91,132),(30,92,129),(31,89,130),(32,90,131),(33,105,67),(34,106,68),(35,107,65),(36,108,66),(37,102,82),(38,103,83),(39,104,84),(40,101,81),(41,52,79),(42,49,80),(43,50,77),(44,51,78),(53,59,95),(54,60,96),(55,57,93),(56,58,94),(69,76,110),(70,73,111),(71,74,112),(72,75,109)], [(1,117,51),(2,118,52),(3,119,49),(4,120,50),(5,64,33),(6,61,34),(7,62,35),(8,63,36),(9,40,69),(10,37,70),(11,38,71),(12,39,72),(13,20,42),(14,17,43),(15,18,44),(16,19,41),(21,55,129),(22,56,130),(23,53,131),(24,54,132),(25,68,133),(26,65,134),(27,66,135),(28,67,136),(29,123,60),(30,124,57),(31,121,58),(32,122,59),(45,103,74),(46,104,75),(47,101,76),(48,102,73),(77,113,86),(78,114,87),(79,115,88),(80,116,85),(81,110,141),(82,111,142),(83,112,143),(84,109,144),(89,126,94),(90,127,95),(91,128,96),(92,125,93),(97,106,140),(98,107,137),(99,108,138),(100,105,139)], [(1,126),(2,127),(3,128),(4,125),(5,11),(6,12),(7,9),(8,10),(13,24),(14,21),(15,22),(16,23),(17,129),(18,130),(19,131),(20,132),(25,75),(26,76),(27,73),(28,74),(29,85),(30,86),(31,87),(32,88),(33,38),(34,39),(35,40),(36,37),(41,53),(42,54),(43,55),(44,56),(45,136),(46,133),(47,134),(48,135),(49,96),(50,93),(51,94),(52,95),(57,77),(58,78),(59,79),(60,80),(61,72),(62,69),(63,70),(64,71),(65,101),(66,102),(67,103),(68,104),(81,107),(82,108),(83,105),(84,106),(89,117),(90,118),(91,119),(92,120),(97,109),(98,110),(99,111),(100,112),(113,124),(114,121),(115,122),(116,123),(137,141),(138,142),(139,143),(140,144)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | ··· | 4L | 4M | ··· | 4X | 6A | ··· | 6L | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4×S3 |
| kernel | C42×C3⋊S3 | C4×C3⋊Dic3 | C122 | C2×C4×C3⋊S3 | C4×C3⋊S3 | C4×C12 | C2×C12 | C12 |
| # reps | 1 | 3 | 1 | 3 | 24 | 4 | 12 | 48 |
Matrix representation of C42×C3⋊S3 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,12] >;
C42×C3⋊S3 in GAP, Magma, Sage, TeX
C_4^2\times C_3\rtimes S_3
% in TeX
G:=Group("C4^2xC3:S3"); // GroupNames label
G:=SmallGroup(288,728);
// by ID
G=gap.SmallGroup(288,728);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations