metabelian, supersoluble, monomial
Aliases: C62.223C23, (C2×C12).205D6, (C22×C6).85D6, C6.97(C4○D12), C12⋊Dic3⋊6C2, C62⋊5C4.6C2, (C6×C12).11C22, C6.93(D4⋊2S3), C6.Dic6⋊19C2, C3⋊6(C23.8D6), (C2×C62).62C22, C32⋊11(C42⋊2C2), C2.9(C12.59D6), C2.7(C12.D6), C23.8(C2×C3⋊S3), (C4×C3⋊Dic3)⋊21C2, C22⋊C4.2(C3⋊S3), (C3×C22⋊C4).10S3, (C3×C6).113(C4○D4), (C2×C6).240(C22×S3), (C32×C22⋊C4).2C2, C22.40(C22×C3⋊S3), (C2×C3⋊Dic3).154C22, (C2×C4).26(C2×C3⋊S3), SmallGroup(288,736)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3 — C62.223C23 |
Generators and relations for C62.223C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=b3, d2=a3, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a3b3c, ede=b3d >
Subgroups: 556 in 180 conjugacy classes, 65 normal (29 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C42⋊2C2, C3⋊Dic3, C3×C12, C62, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×C3⋊Dic3, C6×C12, C2×C62, C23.8D6, C4×C3⋊Dic3, C6.Dic6, C12⋊Dic3, C62⋊5C4, C32×C22⋊C4, C62.223C23
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C42⋊2C2, C2×C3⋊S3, C4○D12, D4⋊2S3, C22×C3⋊S3, C23.8D6, C12.59D6, C12.D6, C62.223C23
►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 32 58 16 39 61)(2 33 59 17 40 62)(3 34 60 18 41 63)(4 35 55 13 42 64)(5 36 56 14 37 65)(6 31 57 15 38 66)(7 29 138 125 24 142)(8 30 133 126 19 143)(9 25 134 121 20 144)(10 26 135 122 21 139)(11 27 136 123 22 140)(12 28 137 124 23 141)(43 71 77 53 94 82)(44 72 78 54 95 83)(45 67 73 49 96 84)(46 68 74 50 91 79)(47 69 75 51 92 80)(48 70 76 52 93 81)(85 118 108 128 111 98)(86 119 103 129 112 99)(87 120 104 130 113 100)(88 115 105 131 114 101)(89 116 106 132 109 102)(90 117 107 127 110 97)
(1 129 16 86)(2 128 17 85)(3 127 18 90)(4 132 13 89)(5 131 14 88)(6 130 15 87)(7 53 125 43)(8 52 126 48)(9 51 121 47)(10 50 122 46)(11 49 123 45)(12 54 124 44)(19 81 30 76)(20 80 25 75)(21 79 26 74)(22 84 27 73)(23 83 28 78)(24 82 29 77)(31 104 38 100)(32 103 39 99)(33 108 40 98)(34 107 41 97)(35 106 42 102)(36 105 37 101)(55 116 64 109)(56 115 65 114)(57 120 66 113)(58 119 61 112)(59 118 62 111)(60 117 63 110)(67 140 96 136)(68 139 91 135)(69 144 92 134)(70 143 93 133)(71 142 94 138)(72 141 95 137)
(1 43 4 46)(2 44 5 47)(3 45 6 48)(7 132 10 129)(8 127 11 130)(9 128 12 131)(13 50 16 53)(14 51 17 54)(15 52 18 49)(19 117 22 120)(20 118 23 115)(21 119 24 116)(25 111 28 114)(26 112 29 109)(27 113 30 110)(31 70 34 67)(32 71 35 68)(33 72 36 69)(37 92 40 95)(38 93 41 96)(39 94 42 91)(55 74 58 77)(56 75 59 78)(57 76 60 73)(61 82 64 79)(62 83 65 80)(63 84 66 81)(85 124 88 121)(86 125 89 122)(87 126 90 123)(97 136 100 133)(98 137 101 134)(99 138 102 135)(103 142 106 139)(104 143 107 140)(105 144 108 141)
(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(43 53)(44 54)(45 49)(46 50)(47 51)(48 52)(67 96)(68 91)(69 92)(70 93)(71 94)(72 95)(73 84)(74 79)(75 80)(76 81)(77 82)(78 83)(85 131)(86 132)(87 127)(88 128)(89 129)(90 130)(97 104)(98 105)(99 106)(100 107)(101 108)(102 103)(109 119)(110 120)(111 115)(112 116)(113 117)(114 118)(121 124)(122 125)(123 126)(133 136)(134 137)(135 138)(139 142)(140 143)(141 144)
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), (1,32,58,16,39,61)(2,33,59,17,40,62)(3,34,60,18,41,63)(4,35,55,13,42,64)(5,36,56,14,37,65)(6,31,57,15,38,66)(7,29,138,125,24,142)(8,30,133,126,19,143)(9,25,134,121,20,144)(10,26,135,122,21,139)(11,27,136,123,22,140)(12,28,137,124,23,141)(43,71,77,53,94,82)(44,72,78,54,95,83)(45,67,73,49,96,84)(46,68,74,50,91,79)(47,69,75,51,92,80)(48,70,76,52,93,81)(85,118,108,128,111,98)(86,119,103,129,112,99)(87,120,104,130,113,100)(88,115,105,131,114,101)(89,116,106,132,109,102)(90,117,107,127,110,97), (1,129,16,86)(2,128,17,85)(3,127,18,90)(4,132,13,89)(5,131,14,88)(6,130,15,87)(7,53,125,43)(8,52,126,48)(9,51,121,47)(10,50,122,46)(11,49,123,45)(12,54,124,44)(19,81,30,76)(20,80,25,75)(21,79,26,74)(22,84,27,73)(23,83,28,78)(24,82,29,77)(31,104,38,100)(32,103,39,99)(33,108,40,98)(34,107,41,97)(35,106,42,102)(36,105,37,101)(55,116,64,109)(56,115,65,114)(57,120,66,113)(58,119,61,112)(59,118,62,111)(60,117,63,110)(67,140,96,136)(68,139,91,135)(69,144,92,134)(70,143,93,133)(71,142,94,138)(72,141,95,137), (1,43,4,46)(2,44,5,47)(3,45,6,48)(7,132,10,129)(8,127,11,130)(9,128,12,131)(13,50,16,53)(14,51,17,54)(15,52,18,49)(19,117,22,120)(20,118,23,115)(21,119,24,116)(25,111,28,114)(26,112,29,109)(27,113,30,110)(31,70,34,67)(32,71,35,68)(33,72,36,69)(37,92,40,95)(38,93,41,96)(39,94,42,91)(55,74,58,77)(56,75,59,78)(57,76,60,73)(61,82,64,79)(62,83,65,80)(63,84,66,81)(85,124,88,121)(86,125,89,122)(87,126,90,123)(97,136,100,133)(98,137,101,134)(99,138,102,135)(103,142,106,139)(104,143,107,140)(105,144,108,141), (7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(43,53)(44,54)(45,49)(46,50)(47,51)(48,52)(67,96)(68,91)(69,92)(70,93)(71,94)(72,95)(73,84)(74,79)(75,80)(76,81)(77,82)(78,83)(85,131)(86,132)(87,127)(88,128)(89,129)(90,130)(97,104)(98,105)(99,106)(100,107)(101,108)(102,103)(109,119)(110,120)(111,115)(112,116)(113,117)(114,118)(121,124)(122,125)(123,126)(133,136)(134,137)(135,138)(139,142)(140,143)(141,144)>;
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), (1,32,58,16,39,61)(2,33,59,17,40,62)(3,34,60,18,41,63)(4,35,55,13,42,64)(5,36,56,14,37,65)(6,31,57,15,38,66)(7,29,138,125,24,142)(8,30,133,126,19,143)(9,25,134,121,20,144)(10,26,135,122,21,139)(11,27,136,123,22,140)(12,28,137,124,23,141)(43,71,77,53,94,82)(44,72,78,54,95,83)(45,67,73,49,96,84)(46,68,74,50,91,79)(47,69,75,51,92,80)(48,70,76,52,93,81)(85,118,108,128,111,98)(86,119,103,129,112,99)(87,120,104,130,113,100)(88,115,105,131,114,101)(89,116,106,132,109,102)(90,117,107,127,110,97), (1,129,16,86)(2,128,17,85)(3,127,18,90)(4,132,13,89)(5,131,14,88)(6,130,15,87)(7,53,125,43)(8,52,126,48)(9,51,121,47)(10,50,122,46)(11,49,123,45)(12,54,124,44)(19,81,30,76)(20,80,25,75)(21,79,26,74)(22,84,27,73)(23,83,28,78)(24,82,29,77)(31,104,38,100)(32,103,39,99)(33,108,40,98)(34,107,41,97)(35,106,42,102)(36,105,37,101)(55,116,64,109)(56,115,65,114)(57,120,66,113)(58,119,61,112)(59,118,62,111)(60,117,63,110)(67,140,96,136)(68,139,91,135)(69,144,92,134)(70,143,93,133)(71,142,94,138)(72,141,95,137), (1,43,4,46)(2,44,5,47)(3,45,6,48)(7,132,10,129)(8,127,11,130)(9,128,12,131)(13,50,16,53)(14,51,17,54)(15,52,18,49)(19,117,22,120)(20,118,23,115)(21,119,24,116)(25,111,28,114)(26,112,29,109)(27,113,30,110)(31,70,34,67)(32,71,35,68)(33,72,36,69)(37,92,40,95)(38,93,41,96)(39,94,42,91)(55,74,58,77)(56,75,59,78)(57,76,60,73)(61,82,64,79)(62,83,65,80)(63,84,66,81)(85,124,88,121)(86,125,89,122)(87,126,90,123)(97,136,100,133)(98,137,101,134)(99,138,102,135)(103,142,106,139)(104,143,107,140)(105,144,108,141), (7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(43,53)(44,54)(45,49)(46,50)(47,51)(48,52)(67,96)(68,91)(69,92)(70,93)(71,94)(72,95)(73,84)(74,79)(75,80)(76,81)(77,82)(78,83)(85,131)(86,132)(87,127)(88,128)(89,129)(90,130)(97,104)(98,105)(99,106)(100,107)(101,108)(102,103)(109,119)(110,120)(111,115)(112,116)(113,117)(114,118)(121,124)(122,125)(123,126)(133,136)(134,137)(135,138)(139,142)(140,143)(141,144) );
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)], [(1,32,58,16,39,61),(2,33,59,17,40,62),(3,34,60,18,41,63),(4,35,55,13,42,64),(5,36,56,14,37,65),(6,31,57,15,38,66),(7,29,138,125,24,142),(8,30,133,126,19,143),(9,25,134,121,20,144),(10,26,135,122,21,139),(11,27,136,123,22,140),(12,28,137,124,23,141),(43,71,77,53,94,82),(44,72,78,54,95,83),(45,67,73,49,96,84),(46,68,74,50,91,79),(47,69,75,51,92,80),(48,70,76,52,93,81),(85,118,108,128,111,98),(86,119,103,129,112,99),(87,120,104,130,113,100),(88,115,105,131,114,101),(89,116,106,132,109,102),(90,117,107,127,110,97)], [(1,129,16,86),(2,128,17,85),(3,127,18,90),(4,132,13,89),(5,131,14,88),(6,130,15,87),(7,53,125,43),(8,52,126,48),(9,51,121,47),(10,50,122,46),(11,49,123,45),(12,54,124,44),(19,81,30,76),(20,80,25,75),(21,79,26,74),(22,84,27,73),(23,83,28,78),(24,82,29,77),(31,104,38,100),(32,103,39,99),(33,108,40,98),(34,107,41,97),(35,106,42,102),(36,105,37,101),(55,116,64,109),(56,115,65,114),(57,120,66,113),(58,119,61,112),(59,118,62,111),(60,117,63,110),(67,140,96,136),(68,139,91,135),(69,144,92,134),(70,143,93,133),(71,142,94,138),(72,141,95,137)], [(1,43,4,46),(2,44,5,47),(3,45,6,48),(7,132,10,129),(8,127,11,130),(9,128,12,131),(13,50,16,53),(14,51,17,54),(15,52,18,49),(19,117,22,120),(20,118,23,115),(21,119,24,116),(25,111,28,114),(26,112,29,109),(27,113,30,110),(31,70,34,67),(32,71,35,68),(33,72,36,69),(37,92,40,95),(38,93,41,96),(39,94,42,91),(55,74,58,77),(56,75,59,78),(57,76,60,73),(61,82,64,79),(62,83,65,80),(63,84,66,81),(85,124,88,121),(86,125,89,122),(87,126,90,123),(97,136,100,133),(98,137,101,134),(99,138,102,135),(103,142,106,139),(104,143,107,140),(105,144,108,141)], [(7,10),(8,11),(9,12),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30),(43,53),(44,54),(45,49),(46,50),(47,51),(48,52),(67,96),(68,91),(69,92),(70,93),(71,94),(72,95),(73,84),(74,79),(75,80),(76,81),(77,82),(78,83),(85,131),(86,132),(87,127),(88,128),(89,129),(90,130),(97,104),(98,105),(99,106),(100,107),(101,108),(102,103),(109,119),(110,120),(111,115),(112,116),(113,117),(114,118),(121,124),(122,125),(123,126),(133,136),(134,137),(135,138),(139,142),(140,143),(141,144)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | ··· | 6L | 6M | ··· | 6T | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 18 | 18 | 18 | 18 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D12 | D4⋊2S3 |
| kernel | C62.223C23 | C4×C3⋊Dic3 | C6.Dic6 | C12⋊Dic3 | C62⋊5C4 | C32×C22⋊C4 | C3×C22⋊C4 | C2×C12 | C22×C6 | C3×C6 | C6 | C6 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 4 | 6 | 16 | 8 |
Matrix representation of C62.223C23 ►in GL6(𝔽13)
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 8 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 10 | 12 |
G:=sub<GL(6,GF(13))| [10,0,0,0,0,0,0,4,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,4,0,0,0,0,0,4],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,8,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,2,0,0,0,0,12,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,10,0,0,0,0,0,12] >;
C62.223C23 in GAP, Magma, Sage, TeX
C_6^2._{223}C_2^3 % in TeX
G:=Group("C6^2.223C2^3"); // GroupNames label
G:=SmallGroup(288,736);
// by ID
G=gap.SmallGroup(288,736);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,219,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=e^2=1,c^2=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^3*b^3*c,e*d*e=b^3*d>;
// generators/relations