metabelian, supersoluble, monomial
Aliases: C62.256C23, (C6×D4)⋊3S3, (C3×C12)⋊13D4, C12⋊4(C3⋊D4), C6.130(S3×D4), C3⋊5(D6⋊3D4), (C2×C12).154D6, C4⋊2(C32⋊7D4), (C22×C6).96D6, C62⋊5C4⋊19C2, C32⋊24(C4⋊D4), C12⋊Dic3⋊23C2, (C6×C12).145C22, (C2×C62).73C22, C6.107(D4⋊2S3), C2.17(C12.D6), (D4×C3×C6)⋊7C2, (C2×C3⋊S3)⋊13D4, C2.26(D4×C3⋊S3), (C2×D4)⋊4(C3⋊S3), (C3×C6).284(C2×D4), C6.125(C2×C3⋊D4), C23.14(C2×C3⋊S3), (C2×C32⋊7D4)⋊11C2, C2.14(C2×C32⋊7D4), (C3×C6).153(C4○D4), (C2×C6).273(C22×S3), C22.60(C22×C3⋊S3), (C22×C3⋊S3).93C22, (C2×C3⋊Dic3).92C22, (C2×C4×C3⋊S3)⋊3C2, (C2×C4).50(C2×C3⋊S3), SmallGroup(288,795)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.256C23
G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=b3, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=a3c, ede=b3d >
Subgroups: 1068 in 282 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4⋊D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C4⋊Dic3, C6.D4, S3×C2×C4, C2×C3⋊D4, C6×D4, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, D4×C32, C22×C3⋊S3, C2×C62, D6⋊3D4, C12⋊Dic3, C62⋊5C4, C2×C4×C3⋊S3, C2×C32⋊7D4, D4×C3×C6, C62.256C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, C3⋊D4, C22×S3, C4⋊D4, C2×C3⋊S3, S3×D4, D4⋊2S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, D6⋊3D4, D4×C3⋊S3, C12.D6, C2×C32⋊7D4, C62.256C23
►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 15 58 33 39 61)(2 16 59 34 40 62)(3 17 60 35 41 63)(4 18 55 36 42 64)(5 13 56 31 37 65)(6 14 57 32 38 66)(7 22 123 116 29 140)(8 23 124 117 30 141)(9 24 125 118 25 142)(10 19 126 119 26 143)(11 20 121 120 27 144)(12 21 122 115 28 139)(43 92 67 73 101 50)(44 93 68 74 102 51)(45 94 69 75 97 52)(46 95 70 76 98 53)(47 96 71 77 99 54)(48 91 72 78 100 49)(79 128 103 109 137 86)(80 129 104 110 138 87)(81 130 105 111 133 88)(82 131 106 112 134 89)(83 132 107 113 135 90)(84 127 108 114 136 85)
(2 6)(3 5)(7 28)(8 27)(9 26)(10 25)(11 30)(12 29)(13 63)(14 62)(15 61)(16 66)(17 65)(18 64)(19 118)(20 117)(21 116)(22 115)(23 120)(24 119)(31 35)(32 34)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 52)(50 51)(53 54)(67 68)(69 72)(70 71)(73 93)(74 92)(75 91)(76 96)(77 95)(78 94)(79 135)(80 134)(81 133)(82 138)(83 137)(84 136)(86 90)(87 89)(103 107)(104 106)(109 132)(110 131)(111 130)(112 129)(113 128)(114 127)(121 124)(122 123)(125 126)(139 140)(141 144)(142 143)
(1 105 33 88)(2 106 34 89)(3 107 35 90)(4 108 36 85)(5 103 31 86)(6 104 32 87)(7 74 116 44)(8 75 117 45)(9 76 118 46)(10 77 119 47)(11 78 120 48)(12 73 115 43)(13 109 37 79)(14 110 38 80)(15 111 39 81)(16 112 40 82)(17 113 41 83)(18 114 42 84)(19 99 26 96)(20 100 27 91)(21 101 28 92)(22 102 29 93)(23 97 30 94)(24 98 25 95)(49 144 72 121)(50 139 67 122)(51 140 68 123)(52 141 69 124)(53 142 70 125)(54 143 71 126)(55 136 64 127)(56 137 65 128)(57 138 66 129)(58 133 61 130)(59 134 62 131)(60 135 63 132)
(1 52)(2 53)(3 54)(4 49)(5 50)(6 51)(7 80)(8 81)(9 82)(10 83)(11 84)(12 79)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 132)(20 127)(21 128)(22 129)(23 130)(24 131)(25 134)(26 135)(27 136)(28 137)(29 138)(30 133)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(85 144)(86 139)(87 140)(88 141)(89 142)(90 143)(103 122)(104 123)(105 124)(106 125)(107 126)(108 121)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)
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), (1,15,58,33,39,61)(2,16,59,34,40,62)(3,17,60,35,41,63)(4,18,55,36,42,64)(5,13,56,31,37,65)(6,14,57,32,38,66)(7,22,123,116,29,140)(8,23,124,117,30,141)(9,24,125,118,25,142)(10,19,126,119,26,143)(11,20,121,120,27,144)(12,21,122,115,28,139)(43,92,67,73,101,50)(44,93,68,74,102,51)(45,94,69,75,97,52)(46,95,70,76,98,53)(47,96,71,77,99,54)(48,91,72,78,100,49)(79,128,103,109,137,86)(80,129,104,110,138,87)(81,130,105,111,133,88)(82,131,106,112,134,89)(83,132,107,113,135,90)(84,127,108,114,136,85), (2,6)(3,5)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,63)(14,62)(15,61)(16,66)(17,65)(18,64)(19,118)(20,117)(21,116)(22,115)(23,120)(24,119)(31,35)(32,34)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,52)(50,51)(53,54)(67,68)(69,72)(70,71)(73,93)(74,92)(75,91)(76,96)(77,95)(78,94)(79,135)(80,134)(81,133)(82,138)(83,137)(84,136)(86,90)(87,89)(103,107)(104,106)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(121,124)(122,123)(125,126)(139,140)(141,144)(142,143), (1,105,33,88)(2,106,34,89)(3,107,35,90)(4,108,36,85)(5,103,31,86)(6,104,32,87)(7,74,116,44)(8,75,117,45)(9,76,118,46)(10,77,119,47)(11,78,120,48)(12,73,115,43)(13,109,37,79)(14,110,38,80)(15,111,39,81)(16,112,40,82)(17,113,41,83)(18,114,42,84)(19,99,26,96)(20,100,27,91)(21,101,28,92)(22,102,29,93)(23,97,30,94)(24,98,25,95)(49,144,72,121)(50,139,67,122)(51,140,68,123)(52,141,69,124)(53,142,70,125)(54,143,71,126)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,133,61,130)(59,134,62,131)(60,135,63,132), (1,52)(2,53)(3,54)(4,49)(5,50)(6,51)(7,80)(8,81)(9,82)(10,83)(11,84)(12,79)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,132)(20,127)(21,128)(22,129)(23,130)(24,131)(25,134)(26,135)(27,136)(28,137)(29,138)(30,133)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(85,144)(86,139)(87,140)(88,141)(89,142)(90,143)(103,122)(104,123)(105,124)(106,125)(107,126)(108,121)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120)>;
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), (1,15,58,33,39,61)(2,16,59,34,40,62)(3,17,60,35,41,63)(4,18,55,36,42,64)(5,13,56,31,37,65)(6,14,57,32,38,66)(7,22,123,116,29,140)(8,23,124,117,30,141)(9,24,125,118,25,142)(10,19,126,119,26,143)(11,20,121,120,27,144)(12,21,122,115,28,139)(43,92,67,73,101,50)(44,93,68,74,102,51)(45,94,69,75,97,52)(46,95,70,76,98,53)(47,96,71,77,99,54)(48,91,72,78,100,49)(79,128,103,109,137,86)(80,129,104,110,138,87)(81,130,105,111,133,88)(82,131,106,112,134,89)(83,132,107,113,135,90)(84,127,108,114,136,85), (2,6)(3,5)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,63)(14,62)(15,61)(16,66)(17,65)(18,64)(19,118)(20,117)(21,116)(22,115)(23,120)(24,119)(31,35)(32,34)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,52)(50,51)(53,54)(67,68)(69,72)(70,71)(73,93)(74,92)(75,91)(76,96)(77,95)(78,94)(79,135)(80,134)(81,133)(82,138)(83,137)(84,136)(86,90)(87,89)(103,107)(104,106)(109,132)(110,131)(111,130)(112,129)(113,128)(114,127)(121,124)(122,123)(125,126)(139,140)(141,144)(142,143), (1,105,33,88)(2,106,34,89)(3,107,35,90)(4,108,36,85)(5,103,31,86)(6,104,32,87)(7,74,116,44)(8,75,117,45)(9,76,118,46)(10,77,119,47)(11,78,120,48)(12,73,115,43)(13,109,37,79)(14,110,38,80)(15,111,39,81)(16,112,40,82)(17,113,41,83)(18,114,42,84)(19,99,26,96)(20,100,27,91)(21,101,28,92)(22,102,29,93)(23,97,30,94)(24,98,25,95)(49,144,72,121)(50,139,67,122)(51,140,68,123)(52,141,69,124)(53,142,70,125)(54,143,71,126)(55,136,64,127)(56,137,65,128)(57,138,66,129)(58,133,61,130)(59,134,62,131)(60,135,63,132), (1,52)(2,53)(3,54)(4,49)(5,50)(6,51)(7,80)(8,81)(9,82)(10,83)(11,84)(12,79)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,132)(20,127)(21,128)(22,129)(23,130)(24,131)(25,134)(26,135)(27,136)(28,137)(29,138)(30,133)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(85,144)(86,139)(87,140)(88,141)(89,142)(90,143)(103,122)(104,123)(105,124)(106,125)(107,126)(108,121)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120) );
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)], [(1,15,58,33,39,61),(2,16,59,34,40,62),(3,17,60,35,41,63),(4,18,55,36,42,64),(5,13,56,31,37,65),(6,14,57,32,38,66),(7,22,123,116,29,140),(8,23,124,117,30,141),(9,24,125,118,25,142),(10,19,126,119,26,143),(11,20,121,120,27,144),(12,21,122,115,28,139),(43,92,67,73,101,50),(44,93,68,74,102,51),(45,94,69,75,97,52),(46,95,70,76,98,53),(47,96,71,77,99,54),(48,91,72,78,100,49),(79,128,103,109,137,86),(80,129,104,110,138,87),(81,130,105,111,133,88),(82,131,106,112,134,89),(83,132,107,113,135,90),(84,127,108,114,136,85)], [(2,6),(3,5),(7,28),(8,27),(9,26),(10,25),(11,30),(12,29),(13,63),(14,62),(15,61),(16,66),(17,65),(18,64),(19,118),(20,117),(21,116),(22,115),(23,120),(24,119),(31,35),(32,34),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,52),(50,51),(53,54),(67,68),(69,72),(70,71),(73,93),(74,92),(75,91),(76,96),(77,95),(78,94),(79,135),(80,134),(81,133),(82,138),(83,137),(84,136),(86,90),(87,89),(103,107),(104,106),(109,132),(110,131),(111,130),(112,129),(113,128),(114,127),(121,124),(122,123),(125,126),(139,140),(141,144),(142,143)], [(1,105,33,88),(2,106,34,89),(3,107,35,90),(4,108,36,85),(5,103,31,86),(6,104,32,87),(7,74,116,44),(8,75,117,45),(9,76,118,46),(10,77,119,47),(11,78,120,48),(12,73,115,43),(13,109,37,79),(14,110,38,80),(15,111,39,81),(16,112,40,82),(17,113,41,83),(18,114,42,84),(19,99,26,96),(20,100,27,91),(21,101,28,92),(22,102,29,93),(23,97,30,94),(24,98,25,95),(49,144,72,121),(50,139,67,122),(51,140,68,123),(52,141,69,124),(53,142,70,125),(54,143,71,126),(55,136,64,127),(56,137,65,128),(57,138,66,129),(58,133,61,130),(59,134,62,131),(60,135,63,132)], [(1,52),(2,53),(3,54),(4,49),(5,50),(6,51),(7,80),(8,81),(9,82),(10,83),(11,84),(12,79),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,132),(20,127),(21,128),(22,129),(23,130),(24,131),(25,134),(26,135),(27,136),(28,137),(29,138),(30,133),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(85,144),(86,139),(87,140),(88,141),(89,142),(90,143),(103,122),(104,123),(105,124),(106,125),(107,126),(108,121),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 6M | ··· | 6AB | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | S3×D4 | D4⋊2S3 |
| kernel | C62.256C23 | C12⋊Dic3 | C62⋊5C4 | C2×C4×C3⋊S3 | C2×C32⋊7D4 | D4×C3×C6 | C6×D4 | C3×C12 | C2×C3⋊S3 | C2×C12 | C22×C6 | C3×C6 | C12 | C6 | C6 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 8 | 2 | 16 | 4 | 4 |
Matrix representation of C62.256C23 ►in GL6(𝔽13)
| 12 | 0 | 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 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 9 | 2 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,11,9,0,0,0,0,4,2] >;
C62.256C23 in GAP, Magma, Sage, TeX
C_6^2._{256}C_2^3 % in TeX
G:=Group("C6^2.256C2^3"); // GroupNames label
G:=SmallGroup(288,795);
// by ID
G=gap.SmallGroup(288,795);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^6=c^2=e^2=1,d^2=b^3,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^3*c,e*d*e=b^3*d>;
// generators/relations