direct product, metabelian, supersoluble, monomial
Aliases: C22×C12⋊S3, C62⋊21D4, C62.276C23, (C2×C6)⋊9D12, C6⋊2(C2×D12), (C2×C12)⋊28D6, (C3×C12)⋊7C23, C12⋊7(C22×S3), C3⋊2(C22×D12), (C6×C12)⋊31C22, (C22×C12)⋊11S3, C6.56(S3×C23), (C3×C6).55C24, C32⋊10(C22×D4), (C22×C6).166D6, (C2×C62).122C22, (C2×C6×C12)⋊9C2, (C3×C6)⋊9(C2×D4), C4⋊2(C22×C3⋊S3), (C23×C3⋊S3)⋊6C2, (C2×C3⋊S3)⋊6C23, C2.4(C23×C3⋊S3), (C22×C4)⋊7(C3⋊S3), C23.40(C2×C3⋊S3), (C22×C3⋊S3)⋊15C22, (C2×C6).285(C22×S3), C22.30(C22×C3⋊S3), (C2×C4)⋊9(C2×C3⋊S3), SmallGroup(288,1005)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 3044 in 708 conjugacy classes, 213 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×4], C4 [×4], C22 [×7], C22 [×32], S3 [×32], C6 [×28], C2×C4 [×6], D4 [×16], C23, C23 [×20], C32, C12 [×16], D6 [×128], C2×C6 [×28], C22×C4, C2×D4 [×12], C24 [×2], C3⋊S3 [×8], C3×C6, C3×C6 [×6], D12 [×64], C2×C12 [×24], C22×S3 [×80], C22×C6 [×4], C22×D4, C3×C12 [×4], C2×C3⋊S3 [×8], C2×C3⋊S3 [×24], C62 [×7], C2×D12 [×48], C22×C12 [×4], S3×C23 [×8], C12⋊S3 [×16], C6×C12 [×6], C22×C3⋊S3 [×12], C22×C3⋊S3 [×8], C2×C62, C22×D12 [×4], C2×C12⋊S3 [×12], C2×C6×C12, C23×C3⋊S3 [×2], C22×C12⋊S3
Quotients:
C1, C2 [×15], C22 [×35], S3 [×4], D4 [×4], C23 [×15], D6 [×28], C2×D4 [×6], C24, C3⋊S3, D12 [×16], C22×S3 [×28], C22×D4, C2×C3⋊S3 [×7], C2×D12 [×24], S3×C23 [×4], C12⋊S3 [×4], C22×C3⋊S3 [×7], C22×D12 [×4], C2×C12⋊S3 [×6], C23×C3⋊S3, C22×C12⋊S3
Generators and relations
G = < a,b,c,d,e | a2=b2=c12=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 >
►On 144 points
Generators in S144
(1 128)(2 129)(3 130)(4 131)(5 132)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 141)(21 142)(22 143)(23 144)(24 133)(25 82)(26 83)(27 84)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 96)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 97)(57 98)(58 99)(59 100)(60 101)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 109)(44 110)(45 111)(46 112)(47 113)(48 114)(49 81)(50 82)(51 83)(52 84)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 85)(69 86)(70 87)(71 88)(72 89)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)(127 133)(128 134)(129 135)(130 136)(131 137)(132 138)
(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)
(1 107 119)(2 108 120)(3 97 109)(4 98 110)(5 99 111)(6 100 112)(7 101 113)(8 102 114)(9 103 115)(10 104 116)(11 105 117)(12 106 118)(13 29 41)(14 30 42)(15 31 43)(16 32 44)(17 33 45)(18 34 46)(19 35 47)(20 36 48)(21 25 37)(22 26 38)(23 27 39)(24 28 40)(49 66 123)(50 67 124)(51 68 125)(52 69 126)(53 70 127)(54 71 128)(55 72 129)(56 61 130)(57 62 131)(58 63 132)(59 64 121)(60 65 122)(73 87 133)(74 88 134)(75 89 135)(76 90 136)(77 91 137)(78 92 138)(79 93 139)(80 94 140)(81 95 141)(82 96 142)(83 85 143)(84 86 144)
(1 131)(2 130)(3 129)(4 128)(5 127)(6 126)(7 125)(8 124)(9 123)(10 122)(11 121)(12 132)(13 137)(14 136)(15 135)(16 134)(17 133)(18 144)(19 143)(20 142)(21 141)(22 140)(23 139)(24 138)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 96)(37 81)(38 80)(39 79)(40 78)(41 77)(42 76)(43 75)(44 74)(45 73)(46 84)(47 83)(48 82)(49 115)(50 114)(51 113)(52 112)(53 111)(54 110)(55 109)(56 120)(57 119)(58 118)(59 117)(60 116)(61 108)(62 107)(63 106)(64 105)(65 104)(66 103)(67 102)(68 101)(69 100)(70 99)(71 98)(72 97)
G:=sub<Sym(144)| (1,128)(2,129)(3,130)(4,131)(5,132)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,141)(21,142)(22,143)(23,144)(24,133)(25,82)(26,83)(27,84)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,96)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,97)(57,98)(58,99)(59,100)(60,101)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,109)(44,110)(45,111)(46,112)(47,113)(48,114)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,85)(69,86)(70,87)(71,88)(72,89)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138), (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), (1,107,119)(2,108,120)(3,97,109)(4,98,110)(5,99,111)(6,100,112)(7,101,113)(8,102,114)(9,103,115)(10,104,116)(11,105,117)(12,106,118)(13,29,41)(14,30,42)(15,31,43)(16,32,44)(17,33,45)(18,34,46)(19,35,47)(20,36,48)(21,25,37)(22,26,38)(23,27,39)(24,28,40)(49,66,123)(50,67,124)(51,68,125)(52,69,126)(53,70,127)(54,71,128)(55,72,129)(56,61,130)(57,62,131)(58,63,132)(59,64,121)(60,65,122)(73,87,133)(74,88,134)(75,89,135)(76,90,136)(77,91,137)(78,92,138)(79,93,139)(80,94,140)(81,95,141)(82,96,142)(83,85,143)(84,86,144), (1,131)(2,130)(3,129)(4,128)(5,127)(6,126)(7,125)(8,124)(9,123)(10,122)(11,121)(12,132)(13,137)(14,136)(15,135)(16,134)(17,133)(18,144)(19,143)(20,142)(21,141)(22,140)(23,139)(24,138)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,96)(37,81)(38,80)(39,79)(40,78)(41,77)(42,76)(43,75)(44,74)(45,73)(46,84)(47,83)(48,82)(49,115)(50,114)(51,113)(52,112)(53,111)(54,110)(55,109)(56,120)(57,119)(58,118)(59,117)(60,116)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)>;
G:=Group( (1,128)(2,129)(3,130)(4,131)(5,132)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,141)(21,142)(22,143)(23,144)(24,133)(25,82)(26,83)(27,84)(28,73)(29,74)(30,75)(31,76)(32,77)(33,78)(34,79)(35,80)(36,81)(37,96)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,91)(45,92)(46,93)(47,94)(48,95)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,97)(57,98)(58,99)(59,100)(60,101)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,109)(44,110)(45,111)(46,112)(47,113)(48,114)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,85)(69,86)(70,87)(71,88)(72,89)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138), (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), (1,107,119)(2,108,120)(3,97,109)(4,98,110)(5,99,111)(6,100,112)(7,101,113)(8,102,114)(9,103,115)(10,104,116)(11,105,117)(12,106,118)(13,29,41)(14,30,42)(15,31,43)(16,32,44)(17,33,45)(18,34,46)(19,35,47)(20,36,48)(21,25,37)(22,26,38)(23,27,39)(24,28,40)(49,66,123)(50,67,124)(51,68,125)(52,69,126)(53,70,127)(54,71,128)(55,72,129)(56,61,130)(57,62,131)(58,63,132)(59,64,121)(60,65,122)(73,87,133)(74,88,134)(75,89,135)(76,90,136)(77,91,137)(78,92,138)(79,93,139)(80,94,140)(81,95,141)(82,96,142)(83,85,143)(84,86,144), (1,131)(2,130)(3,129)(4,128)(5,127)(6,126)(7,125)(8,124)(9,123)(10,122)(11,121)(12,132)(13,137)(14,136)(15,135)(16,134)(17,133)(18,144)(19,143)(20,142)(21,141)(22,140)(23,139)(24,138)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,96)(37,81)(38,80)(39,79)(40,78)(41,77)(42,76)(43,75)(44,74)(45,73)(46,84)(47,83)(48,82)(49,115)(50,114)(51,113)(52,112)(53,111)(54,110)(55,109)(56,120)(57,119)(58,118)(59,117)(60,116)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97) );
G=PermutationGroup([(1,128),(2,129),(3,130),(4,131),(5,132),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,127),(13,134),(14,135),(15,136),(16,137),(17,138),(18,139),(19,140),(20,141),(21,142),(22,143),(23,144),(24,133),(25,82),(26,83),(27,84),(28,73),(29,74),(30,75),(31,76),(32,77),(33,78),(34,79),(35,80),(36,81),(37,96),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,91),(45,92),(46,93),(47,94),(48,95),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(55,108),(56,97),(57,98),(58,99),(59,100),(60,101),(61,109),(62,110),(63,111),(64,112),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,97),(32,98),(33,99),(34,100),(35,101),(36,102),(37,115),(38,116),(39,117),(40,118),(41,119),(42,120),(43,109),(44,110),(45,111),(46,112),(47,113),(48,114),(49,81),(50,82),(51,83),(52,84),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,90),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,85),(69,86),(70,87),(71,88),(72,89),(121,139),(122,140),(123,141),(124,142),(125,143),(126,144),(127,133),(128,134),(129,135),(130,136),(131,137),(132,138)], [(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)], [(1,107,119),(2,108,120),(3,97,109),(4,98,110),(5,99,111),(6,100,112),(7,101,113),(8,102,114),(9,103,115),(10,104,116),(11,105,117),(12,106,118),(13,29,41),(14,30,42),(15,31,43),(16,32,44),(17,33,45),(18,34,46),(19,35,47),(20,36,48),(21,25,37),(22,26,38),(23,27,39),(24,28,40),(49,66,123),(50,67,124),(51,68,125),(52,69,126),(53,70,127),(54,71,128),(55,72,129),(56,61,130),(57,62,131),(58,63,132),(59,64,121),(60,65,122),(73,87,133),(74,88,134),(75,89,135),(76,90,136),(77,91,137),(78,92,138),(79,93,139),(80,94,140),(81,95,141),(82,96,142),(83,85,143),(84,86,144)], [(1,131),(2,130),(3,129),(4,128),(5,127),(6,126),(7,125),(8,124),(9,123),(10,122),(11,121),(12,132),(13,137),(14,136),(15,135),(16,134),(17,133),(18,144),(19,143),(20,142),(21,141),(22,140),(23,139),(24,138),(25,95),(26,94),(27,93),(28,92),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,96),(37,81),(38,80),(39,79),(40,78),(41,77),(42,76),(43,75),(44,74),(45,73),(46,84),(47,83),(48,82),(49,115),(50,114),(51,113),(52,112),(53,111),(54,110),(55,109),(56,120),(57,119),(58,118),(59,117),(60,116),(61,108),(62,107),(63,106),(64,105),(65,104),(66,103),(67,102),(68,101),(69,100),(70,99),(71,98),(72,97)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 12 |
| 12 | 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 7 | 3 | 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 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,3,0,0,0,0,10,7,0,0,0,0,0,0,12,1,0,0,0,0,0,1] >;
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6AB | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D12 |
| kernel | C22×C12⋊S3 | C2×C12⋊S3 | C2×C6×C12 | C23×C3⋊S3 | C22×C12 | C62 | C2×C12 | C22×C6 | C2×C6 |
| # reps | 1 | 12 | 1 | 2 | 4 | 4 | 24 | 4 | 32 |
In GAP, Magma, Sage, TeX
C_2^2\times C_{12}\rtimes S_3 % in TeX
G:=Group("C2^2xC12:S3"); // GroupNames label
G:=SmallGroup(288,1005);
// by ID
G=gap.SmallGroup(288,1005);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^12=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
×
⋊
ℤ
𝔽
○
≀