direct product, metabelian, supersoluble, monomial
Aliases: C22×C12⋊S3, C62⋊21D4, C62.276C23, C6⋊2(C2×D12), (C2×C6)⋊9D12, (C2×C12)⋊28D6, C12⋊7(C22×S3), (C3×C12)⋊7C23, 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
Generators and relations for C22×C12⋊S3
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 >
Subgroups: 3044 in 708 conjugacy classes, 213 normal (9 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C23, C32, C12, D6, C2×C6, C22×C4, C2×D4, C24, C3⋊S3, C3×C6, C3×C6, D12, C2×C12, C22×S3, C22×C6, C22×D4, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, C22×C12, S3×C23, C12⋊S3, C6×C12, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C22×D12, C2×C12⋊S3, C2×C6×C12, C23×C3⋊S3, C22×C12⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊S3, D12, C22×S3, C22×D4, C2×C3⋊S3, C2×D12, S3×C23, C12⋊S3, C22×C3⋊S3, C22×D12, C2×C12⋊S3, C23×C3⋊S3, C22×C12⋊S3
►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 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 97)(60 98)(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 78)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 73)(57 74)(58 75)(59 76)(60 77)(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)
(1 105 119)(2 106 120)(3 107 109)(4 108 110)(5 97 111)(6 98 112)(7 99 113)(8 100 114)(9 101 115)(10 102 116)(11 103 117)(12 104 118)(13 27 41)(14 28 42)(15 29 43)(16 30 44)(17 31 45)(18 32 46)(19 33 47)(20 34 48)(21 35 37)(22 36 38)(23 25 39)(24 26 40)(49 65 122)(50 66 123)(51 67 124)(52 68 125)(53 69 126)(54 70 127)(55 71 128)(56 72 129)(57 61 130)(58 62 131)(59 63 132)(60 64 121)(73 89 135)(74 90 136)(75 91 137)(76 92 138)(77 93 139)(78 94 140)(79 95 141)(80 96 142)(81 85 143)(82 86 144)(83 87 133)(84 88 134)
(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 93)(26 92)(27 91)(28 90)(29 89)(30 88)(31 87)(32 86)(33 85)(34 96)(35 95)(36 94)(37 79)(38 78)(39 77)(40 76)(41 75)(42 74)(43 73)(44 84)(45 83)(46 82)(47 81)(48 80)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 120)(58 119)(59 118)(60 117)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 108)(72 107)
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)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,97)(60,98)(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)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,73)(57,74)(58,75)(59,76)(60,77)(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), (1,105,119)(2,106,120)(3,107,109)(4,108,110)(5,97,111)(6,98,112)(7,99,113)(8,100,114)(9,101,115)(10,102,116)(11,103,117)(12,104,118)(13,27,41)(14,28,42)(15,29,43)(16,30,44)(17,31,45)(18,32,46)(19,33,47)(20,34,48)(21,35,37)(22,36,38)(23,25,39)(24,26,40)(49,65,122)(50,66,123)(51,67,124)(52,68,125)(53,69,126)(54,70,127)(55,71,128)(56,72,129)(57,61,130)(58,62,131)(59,63,132)(60,64,121)(73,89,135)(74,90,136)(75,91,137)(76,92,138)(77,93,139)(78,94,140)(79,95,141)(80,96,142)(81,85,143)(82,86,144)(83,87,133)(84,88,134), (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)(25,93)(26,92)(27,91)(28,90)(29,89)(30,88)(31,87)(32,86)(33,85)(34,96)(35,95)(36,94)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(43,73)(44,84)(45,83)(46,82)(47,81)(48,80)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,120)(58,119)(59,118)(60,117)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,108)(72,107)>;
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)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,97)(60,98)(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)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,73)(57,74)(58,75)(59,76)(60,77)(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), (1,105,119)(2,106,120)(3,107,109)(4,108,110)(5,97,111)(6,98,112)(7,99,113)(8,100,114)(9,101,115)(10,102,116)(11,103,117)(12,104,118)(13,27,41)(14,28,42)(15,29,43)(16,30,44)(17,31,45)(18,32,46)(19,33,47)(20,34,48)(21,35,37)(22,36,38)(23,25,39)(24,26,40)(49,65,122)(50,66,123)(51,67,124)(52,68,125)(53,69,126)(54,70,127)(55,71,128)(56,72,129)(57,61,130)(58,62,131)(59,63,132)(60,64,121)(73,89,135)(74,90,136)(75,91,137)(76,92,138)(77,93,139)(78,94,140)(79,95,141)(80,96,142)(81,85,143)(82,86,144)(83,87,133)(84,88,134), (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)(25,93)(26,92)(27,91)(28,90)(29,89)(30,88)(31,87)(32,86)(33,85)(34,96)(35,95)(36,94)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(43,73)(44,84)(45,83)(46,82)(47,81)(48,80)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,120)(58,119)(59,118)(60,117)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,108)(72,107) );
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),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,97),(60,98),(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),(49,78),(50,79),(51,80),(52,81),(53,82),(54,83),(55,84),(56,73),(57,74),(58,75),(59,76),(60,77),(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)], [(1,105,119),(2,106,120),(3,107,109),(4,108,110),(5,97,111),(6,98,112),(7,99,113),(8,100,114),(9,101,115),(10,102,116),(11,103,117),(12,104,118),(13,27,41),(14,28,42),(15,29,43),(16,30,44),(17,31,45),(18,32,46),(19,33,47),(20,34,48),(21,35,37),(22,36,38),(23,25,39),(24,26,40),(49,65,122),(50,66,123),(51,67,124),(52,68,125),(53,69,126),(54,70,127),(55,71,128),(56,72,129),(57,61,130),(58,62,131),(59,63,132),(60,64,121),(73,89,135),(74,90,136),(75,91,137),(76,92,138),(77,93,139),(78,94,140),(79,95,141),(80,96,142),(81,85,143),(82,86,144),(83,87,133),(84,88,134)], [(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),(25,93),(26,92),(27,91),(28,90),(29,89),(30,88),(31,87),(32,86),(33,85),(34,96),(35,95),(36,94),(37,79),(38,78),(39,77),(40,76),(41,75),(42,74),(43,73),(44,84),(45,83),(46,82),(47,81),(48,80),(49,116),(50,115),(51,114),(52,113),(53,112),(54,111),(55,110),(56,109),(57,120),(58,119),(59,118),(60,117),(61,106),(62,105),(63,104),(64,103),(65,102),(66,101),(67,100),(68,99),(69,98),(70,97),(71,108),(72,107)]])
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 |
Matrix representation of C22×C12⋊S3 ►in 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] >;
C22×C12⋊S3 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