metabelian, supersoluble, monomial
Aliases: C62.254C23, (C6×D4).17S3, (C3×C12).100D4, (C2×C12).153D6, (C22×C6).95D6, C62⋊5C4⋊17C2, C12.59(C3⋊D4), C4.7(C32⋊7D4), C3⋊4(C23.12D6), (C6×C12).144C22, C32⋊16(C4.4D4), (C2×C62).72C22, C6.106(D4⋊2S3), C2.16(C12.D6), (D4×C3×C6).10C2, (C4×C3⋊Dic3)⋊9C2, (C2×D4).6(C3⋊S3), (C3×C6).282(C2×D4), C6.123(C2×C3⋊D4), C23.13(C2×C3⋊S3), (C2×C32⋊4Q8)⋊16C2, C2.12(C2×C32⋊7D4), (C3×C6).152(C4○D4), (C2×C6).271(C22×S3), C22.58(C22×C3⋊S3), (C2×C3⋊Dic3).166C22, (C2×C4).49(C2×C3⋊S3), SmallGroup(288,793)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C4×C3⋊Dic3 — C62.254C23 |
Generators and relations for C62.254C23
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=a3d >
Subgroups: 716 in 228 conjugacy classes, 77 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3×C6, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, C4.4D4, C3⋊Dic3, C3×C12, C62, C62, C4×Dic3, C6.D4, C2×Dic6, C6×D4, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, D4×C32, C2×C62, C23.12D6, C4×C3⋊Dic3, C62⋊5C4, C2×C32⋊4Q8, D4×C3×C6, C62.254C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, C3⋊D4, C22×S3, C4.4D4, C2×C3⋊S3, D4⋊2S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, C23.12D6, C12.D6, C2×C32⋊7D4, C62.254C23
►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 31 38 57 16 52)(2 32 39 58 17 53)(3 33 40 59 18 54)(4 34 41 60 13 49)(5 35 42 55 14 50)(6 36 37 56 15 51)(7 142 27 22 123 137)(8 143 28 23 124 138)(9 144 29 24 125 133)(10 139 30 19 126 134)(11 140 25 20 121 135)(12 141 26 21 122 136)(43 93 73 63 83 71)(44 94 74 64 84 72)(45 95 75 65 79 67)(46 96 76 66 80 68)(47 91 77 61 81 69)(48 92 78 62 82 70)(85 99 112 104 132 118)(86 100 113 105 127 119)(87 101 114 106 128 120)(88 102 109 107 129 115)(89 97 110 108 130 116)(90 98 111 103 131 117)
(1 100 57 127)(2 99 58 132)(3 98 59 131)(4 97 60 130)(5 102 55 129)(6 101 56 128)(7 94 22 84)(8 93 23 83)(9 92 24 82)(10 91 19 81)(11 96 20 80)(12 95 21 79)(13 108 34 89)(14 107 35 88)(15 106 36 87)(16 105 31 86)(17 104 32 85)(18 103 33 90)(25 68 135 76)(26 67 136 75)(27 72 137 74)(28 71 138 73)(29 70 133 78)(30 69 134 77)(37 120 51 114)(38 119 52 113)(39 118 53 112)(40 117 54 111)(41 116 49 110)(42 115 50 109)(43 124 63 143)(44 123 64 142)(45 122 65 141)(46 121 66 140)(47 126 61 139)(48 125 62 144)
(1 67 4 70)(2 68 5 71)(3 69 6 72)(7 111 10 114)(8 112 11 109)(9 113 12 110)(13 62 16 65)(14 63 17 66)(15 64 18 61)(19 120 22 117)(20 115 23 118)(21 116 24 119)(25 129 28 132)(26 130 29 127)(27 131 30 128)(31 45 34 48)(32 46 35 43)(33 47 36 44)(37 94 40 91)(38 95 41 92)(39 96 42 93)(49 82 52 79)(50 83 53 80)(51 84 54 81)(55 73 58 76)(56 74 59 77)(57 75 60 78)(85 121 88 124)(86 122 89 125)(87 123 90 126)(97 133 100 136)(98 134 101 137)(99 135 102 138)(103 139 106 142)(104 140 107 143)(105 141 108 144)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 117)(8 118)(9 119)(10 120)(11 115)(12 116)(13 62)(14 63)(15 64)(16 65)(17 66)(18 61)(19 114)(20 109)(21 110)(22 111)(23 112)(24 113)(25 102)(26 97)(27 98)(28 99)(29 100)(30 101)(31 45)(32 46)(33 47)(34 48)(35 43)(36 44)(37 94)(38 95)(39 96)(40 91)(41 92)(42 93)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(85 143)(86 144)(87 139)(88 140)(89 141)(90 142)(103 123)(104 124)(105 125)(106 126)(107 121)(108 122)(127 133)(128 134)(129 135)(130 136)(131 137)(132 138)
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), (1,31,38,57,16,52)(2,32,39,58,17,53)(3,33,40,59,18,54)(4,34,41,60,13,49)(5,35,42,55,14,50)(6,36,37,56,15,51)(7,142,27,22,123,137)(8,143,28,23,124,138)(9,144,29,24,125,133)(10,139,30,19,126,134)(11,140,25,20,121,135)(12,141,26,21,122,136)(43,93,73,63,83,71)(44,94,74,64,84,72)(45,95,75,65,79,67)(46,96,76,66,80,68)(47,91,77,61,81,69)(48,92,78,62,82,70)(85,99,112,104,132,118)(86,100,113,105,127,119)(87,101,114,106,128,120)(88,102,109,107,129,115)(89,97,110,108,130,116)(90,98,111,103,131,117), (1,100,57,127)(2,99,58,132)(3,98,59,131)(4,97,60,130)(5,102,55,129)(6,101,56,128)(7,94,22,84)(8,93,23,83)(9,92,24,82)(10,91,19,81)(11,96,20,80)(12,95,21,79)(13,108,34,89)(14,107,35,88)(15,106,36,87)(16,105,31,86)(17,104,32,85)(18,103,33,90)(25,68,135,76)(26,67,136,75)(27,72,137,74)(28,71,138,73)(29,70,133,78)(30,69,134,77)(37,120,51,114)(38,119,52,113)(39,118,53,112)(40,117,54,111)(41,116,49,110)(42,115,50,109)(43,124,63,143)(44,123,64,142)(45,122,65,141)(46,121,66,140)(47,126,61,139)(48,125,62,144), (1,67,4,70)(2,68,5,71)(3,69,6,72)(7,111,10,114)(8,112,11,109)(9,113,12,110)(13,62,16,65)(14,63,17,66)(15,64,18,61)(19,120,22,117)(20,115,23,118)(21,116,24,119)(25,129,28,132)(26,130,29,127)(27,131,30,128)(31,45,34,48)(32,46,35,43)(33,47,36,44)(37,94,40,91)(38,95,41,92)(39,96,42,93)(49,82,52,79)(50,83,53,80)(51,84,54,81)(55,73,58,76)(56,74,59,77)(57,75,60,78)(85,121,88,124)(86,122,89,125)(87,123,90,126)(97,133,100,136)(98,134,101,137)(99,135,102,138)(103,139,106,142)(104,140,107,143)(105,141,108,144), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,117)(8,118)(9,119)(10,120)(11,115)(12,116)(13,62)(14,63)(15,64)(16,65)(17,66)(18,61)(19,114)(20,109)(21,110)(22,111)(23,112)(24,113)(25,102)(26,97)(27,98)(28,99)(29,100)(30,101)(31,45)(32,46)(33,47)(34,48)(35,43)(36,44)(37,94)(38,95)(39,96)(40,91)(41,92)(42,93)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(85,143)(86,144)(87,139)(88,140)(89,141)(90,142)(103,123)(104,124)(105,125)(106,126)(107,121)(108,122)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138)>;
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), (1,31,38,57,16,52)(2,32,39,58,17,53)(3,33,40,59,18,54)(4,34,41,60,13,49)(5,35,42,55,14,50)(6,36,37,56,15,51)(7,142,27,22,123,137)(8,143,28,23,124,138)(9,144,29,24,125,133)(10,139,30,19,126,134)(11,140,25,20,121,135)(12,141,26,21,122,136)(43,93,73,63,83,71)(44,94,74,64,84,72)(45,95,75,65,79,67)(46,96,76,66,80,68)(47,91,77,61,81,69)(48,92,78,62,82,70)(85,99,112,104,132,118)(86,100,113,105,127,119)(87,101,114,106,128,120)(88,102,109,107,129,115)(89,97,110,108,130,116)(90,98,111,103,131,117), (1,100,57,127)(2,99,58,132)(3,98,59,131)(4,97,60,130)(5,102,55,129)(6,101,56,128)(7,94,22,84)(8,93,23,83)(9,92,24,82)(10,91,19,81)(11,96,20,80)(12,95,21,79)(13,108,34,89)(14,107,35,88)(15,106,36,87)(16,105,31,86)(17,104,32,85)(18,103,33,90)(25,68,135,76)(26,67,136,75)(27,72,137,74)(28,71,138,73)(29,70,133,78)(30,69,134,77)(37,120,51,114)(38,119,52,113)(39,118,53,112)(40,117,54,111)(41,116,49,110)(42,115,50,109)(43,124,63,143)(44,123,64,142)(45,122,65,141)(46,121,66,140)(47,126,61,139)(48,125,62,144), (1,67,4,70)(2,68,5,71)(3,69,6,72)(7,111,10,114)(8,112,11,109)(9,113,12,110)(13,62,16,65)(14,63,17,66)(15,64,18,61)(19,120,22,117)(20,115,23,118)(21,116,24,119)(25,129,28,132)(26,130,29,127)(27,131,30,128)(31,45,34,48)(32,46,35,43)(33,47,36,44)(37,94,40,91)(38,95,41,92)(39,96,42,93)(49,82,52,79)(50,83,53,80)(51,84,54,81)(55,73,58,76)(56,74,59,77)(57,75,60,78)(85,121,88,124)(86,122,89,125)(87,123,90,126)(97,133,100,136)(98,134,101,137)(99,135,102,138)(103,139,106,142)(104,140,107,143)(105,141,108,144), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,117)(8,118)(9,119)(10,120)(11,115)(12,116)(13,62)(14,63)(15,64)(16,65)(17,66)(18,61)(19,114)(20,109)(21,110)(22,111)(23,112)(24,113)(25,102)(26,97)(27,98)(28,99)(29,100)(30,101)(31,45)(32,46)(33,47)(34,48)(35,43)(36,44)(37,94)(38,95)(39,96)(40,91)(41,92)(42,93)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(85,143)(86,144)(87,139)(88,140)(89,141)(90,142)(103,123)(104,124)(105,125)(106,126)(107,121)(108,122)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138) );
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)], [(1,31,38,57,16,52),(2,32,39,58,17,53),(3,33,40,59,18,54),(4,34,41,60,13,49),(5,35,42,55,14,50),(6,36,37,56,15,51),(7,142,27,22,123,137),(8,143,28,23,124,138),(9,144,29,24,125,133),(10,139,30,19,126,134),(11,140,25,20,121,135),(12,141,26,21,122,136),(43,93,73,63,83,71),(44,94,74,64,84,72),(45,95,75,65,79,67),(46,96,76,66,80,68),(47,91,77,61,81,69),(48,92,78,62,82,70),(85,99,112,104,132,118),(86,100,113,105,127,119),(87,101,114,106,128,120),(88,102,109,107,129,115),(89,97,110,108,130,116),(90,98,111,103,131,117)], [(1,100,57,127),(2,99,58,132),(3,98,59,131),(4,97,60,130),(5,102,55,129),(6,101,56,128),(7,94,22,84),(8,93,23,83),(9,92,24,82),(10,91,19,81),(11,96,20,80),(12,95,21,79),(13,108,34,89),(14,107,35,88),(15,106,36,87),(16,105,31,86),(17,104,32,85),(18,103,33,90),(25,68,135,76),(26,67,136,75),(27,72,137,74),(28,71,138,73),(29,70,133,78),(30,69,134,77),(37,120,51,114),(38,119,52,113),(39,118,53,112),(40,117,54,111),(41,116,49,110),(42,115,50,109),(43,124,63,143),(44,123,64,142),(45,122,65,141),(46,121,66,140),(47,126,61,139),(48,125,62,144)], [(1,67,4,70),(2,68,5,71),(3,69,6,72),(7,111,10,114),(8,112,11,109),(9,113,12,110),(13,62,16,65),(14,63,17,66),(15,64,18,61),(19,120,22,117),(20,115,23,118),(21,116,24,119),(25,129,28,132),(26,130,29,127),(27,131,30,128),(31,45,34,48),(32,46,35,43),(33,47,36,44),(37,94,40,91),(38,95,41,92),(39,96,42,93),(49,82,52,79),(50,83,53,80),(51,84,54,81),(55,73,58,76),(56,74,59,77),(57,75,60,78),(85,121,88,124),(86,122,89,125),(87,123,90,126),(97,133,100,136),(98,134,101,137),(99,135,102,138),(103,139,106,142),(104,140,107,143),(105,141,108,144)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,117),(8,118),(9,119),(10,120),(11,115),(12,116),(13,62),(14,63),(15,64),(16,65),(17,66),(18,61),(19,114),(20,109),(21,110),(22,111),(23,112),(24,113),(25,102),(26,97),(27,98),(28,99),(29,100),(30,101),(31,45),(32,46),(33,47),(34,48),(35,43),(36,44),(37,94),(38,95),(39,96),(40,91),(41,92),(42,93),(49,82),(50,83),(51,84),(52,79),(53,80),(54,81),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(85,143),(86,144),(87,139),(88,140),(89,141),(90,142),(103,123),(104,124),(105,125),(106,126),(107,121),(108,122),(127,133),(128,134),(129,135),(130,136),(131,137),(132,138)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6L | 6M | ··· | 6AB | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | D4⋊2S3 |
| kernel | C62.254C23 | C4×C3⋊Dic3 | C62⋊5C4 | C2×C32⋊4Q8 | D4×C3×C6 | C6×D4 | C3×C12 | C2×C12 | C22×C6 | C3×C6 | C12 | C6 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 4 | 8 | 4 | 16 | 8 |
Matrix representation of C62.254C23 ►in GL6(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 10 |
| 0 | 0 | 0 | 0 | 8 | 5 |
| 1 | 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 | 12 | 2 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [9,9,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,10,0,0,0,0,5,1,0,0,0,0,0,0,3,7,0,0,0,0,10,10,0,0,0,0,0,0,8,8,0,0,0,0,10,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,2,1],[1,3,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,11,12] >;
C62.254C23 in GAP, Magma, Sage, TeX
C_6^2._{254}C_2^3 % in TeX
G:=Group("C6^2.254C2^3"); // GroupNames label
G:=SmallGroup(288,793);
// by ID
G=gap.SmallGroup(288,793);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,135,2693,9414]);
// 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=a^3*d>;
// generators/relations