metabelian, supersoluble, monomial
Aliases: C62.80D6, C33⋊12(C4⋊C4), (C3×C6).43D12, Dic3⋊(C3⋊Dic3), (C32×C6).6Q8, C6.31(S3×Dic3), (C3×C6).16Dic6, (C32×C6).44D4, (C3×Dic3)⋊1Dic3, C32⋊7(C4⋊Dic3), (C6×Dic3).11S3, (C32×Dic3)⋊3C4, C6.5(C32⋊2Q8), C6.18(C12⋊S3), C2.3(C33⋊8D4), C3⋊1(Dic3⋊Dic3), C6.13(C3⋊D12), C3⋊1(C12⋊Dic3), C2.1(C33⋊4Q8), C6.1(C32⋊4Q8), (C3×C62).10C22, C32⋊13(Dic3⋊C4), (C2×C6).34S32, (C3×C6).92(C4×S3), C22.8(S3×C3⋊S3), C6.5(C2×C3⋊Dic3), C2.5(S3×C3⋊Dic3), (Dic3×C3×C6).7C2, (C6×C3⋊Dic3).3C2, (C2×C3⋊Dic3).5S3, (C3×C6).81(C3⋊D4), (C32×C6).41(C2×C4), (C2×C33⋊5C4).2C2, (C3×C6).39(C2×Dic3), (C2×Dic3).1(C3⋊S3), (C2×C6).16(C2×C3⋊S3), SmallGroup(432,452)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.80D6
G = < a,b,c,d | a6=b6=1, c6=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c5 >
Subgroups: 984 in 220 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C4⋊Dic3, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C33⋊5C4, C3×C62, Dic3⋊Dic3, C12⋊Dic3, Dic3×C3×C6, C6×C3⋊Dic3, C2×C33⋊5C4, C62.80D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, C3⋊S3, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊Dic3, S32, C2×C3⋊S3, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C32⋊2Q8, C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, S3×C3⋊S3, Dic3⋊Dic3, C12⋊Dic3, S3×C3⋊Dic3, C33⋊8D4, C33⋊4Q8, C62.80D6
►On 144 points
Generators in S144
(1 121 57 115 13 79)(2 122 58 116 14 80)(3 123 59 117 15 81)(4 124 60 118 16 82)(5 125 49 119 17 83)(6 126 50 120 18 84)(7 127 51 109 19 73)(8 128 52 110 20 74)(9 129 53 111 21 75)(10 130 54 112 22 76)(11 131 55 113 23 77)(12 132 56 114 24 78)(25 94 40 100 62 140)(26 95 41 101 63 141)(27 96 42 102 64 142)(28 85 43 103 65 143)(29 86 44 104 66 144)(30 87 45 105 67 133)(31 88 46 106 68 134)(32 89 47 107 69 135)(33 90 48 108 70 136)(34 91 37 97 71 137)(35 92 38 98 72 138)(36 93 39 99 61 139)
(1 55 21 7 49 15)(2 16 50 8 22 56)(3 57 23 9 51 17)(4 18 52 10 24 58)(5 59 13 11 53 19)(6 20 54 12 14 60)(25 64 44 31 70 38)(26 39 71 32 45 65)(27 66 46 33 72 40)(28 41 61 34 47 67)(29 68 48 35 62 42)(30 43 63 36 37 69)(73 125 117 79 131 111)(74 112 132 80 118 126)(75 127 119 81 121 113)(76 114 122 82 120 128)(77 129 109 83 123 115)(78 116 124 84 110 130)(85 101 139 91 107 133)(86 134 108 92 140 102)(87 103 141 93 97 135)(88 136 98 94 142 104)(89 105 143 95 99 137)(90 138 100 96 144 106)
(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 30 115 105)(2 29 116 104)(3 28 117 103)(4 27 118 102)(5 26 119 101)(6 25 120 100)(7 36 109 99)(8 35 110 98)(9 34 111 97)(10 33 112 108)(11 32 113 107)(12 31 114 106)(13 45 121 133)(14 44 122 144)(15 43 123 143)(16 42 124 142)(17 41 125 141)(18 40 126 140)(19 39 127 139)(20 38 128 138)(21 37 129 137)(22 48 130 136)(23 47 131 135)(24 46 132 134)(49 63 83 95)(50 62 84 94)(51 61 73 93)(52 72 74 92)(53 71 75 91)(54 70 76 90)(55 69 77 89)(56 68 78 88)(57 67 79 87)(58 66 80 86)(59 65 81 85)(60 64 82 96)
G:=sub<Sym(144)| (1,121,57,115,13,79)(2,122,58,116,14,80)(3,123,59,117,15,81)(4,124,60,118,16,82)(5,125,49,119,17,83)(6,126,50,120,18,84)(7,127,51,109,19,73)(8,128,52,110,20,74)(9,129,53,111,21,75)(10,130,54,112,22,76)(11,131,55,113,23,77)(12,132,56,114,24,78)(25,94,40,100,62,140)(26,95,41,101,63,141)(27,96,42,102,64,142)(28,85,43,103,65,143)(29,86,44,104,66,144)(30,87,45,105,67,133)(31,88,46,106,68,134)(32,89,47,107,69,135)(33,90,48,108,70,136)(34,91,37,97,71,137)(35,92,38,98,72,138)(36,93,39,99,61,139), (1,55,21,7,49,15)(2,16,50,8,22,56)(3,57,23,9,51,17)(4,18,52,10,24,58)(5,59,13,11,53,19)(6,20,54,12,14,60)(25,64,44,31,70,38)(26,39,71,32,45,65)(27,66,46,33,72,40)(28,41,61,34,47,67)(29,68,48,35,62,42)(30,43,63,36,37,69)(73,125,117,79,131,111)(74,112,132,80,118,126)(75,127,119,81,121,113)(76,114,122,82,120,128)(77,129,109,83,123,115)(78,116,124,84,110,130)(85,101,139,91,107,133)(86,134,108,92,140,102)(87,103,141,93,97,135)(88,136,98,94,142,104)(89,105,143,95,99,137)(90,138,100,96,144,106), (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), (1,30,115,105)(2,29,116,104)(3,28,117,103)(4,27,118,102)(5,26,119,101)(6,25,120,100)(7,36,109,99)(8,35,110,98)(9,34,111,97)(10,33,112,108)(11,32,113,107)(12,31,114,106)(13,45,121,133)(14,44,122,144)(15,43,123,143)(16,42,124,142)(17,41,125,141)(18,40,126,140)(19,39,127,139)(20,38,128,138)(21,37,129,137)(22,48,130,136)(23,47,131,135)(24,46,132,134)(49,63,83,95)(50,62,84,94)(51,61,73,93)(52,72,74,92)(53,71,75,91)(54,70,76,90)(55,69,77,89)(56,68,78,88)(57,67,79,87)(58,66,80,86)(59,65,81,85)(60,64,82,96)>;
G:=Group( (1,121,57,115,13,79)(2,122,58,116,14,80)(3,123,59,117,15,81)(4,124,60,118,16,82)(5,125,49,119,17,83)(6,126,50,120,18,84)(7,127,51,109,19,73)(8,128,52,110,20,74)(9,129,53,111,21,75)(10,130,54,112,22,76)(11,131,55,113,23,77)(12,132,56,114,24,78)(25,94,40,100,62,140)(26,95,41,101,63,141)(27,96,42,102,64,142)(28,85,43,103,65,143)(29,86,44,104,66,144)(30,87,45,105,67,133)(31,88,46,106,68,134)(32,89,47,107,69,135)(33,90,48,108,70,136)(34,91,37,97,71,137)(35,92,38,98,72,138)(36,93,39,99,61,139), (1,55,21,7,49,15)(2,16,50,8,22,56)(3,57,23,9,51,17)(4,18,52,10,24,58)(5,59,13,11,53,19)(6,20,54,12,14,60)(25,64,44,31,70,38)(26,39,71,32,45,65)(27,66,46,33,72,40)(28,41,61,34,47,67)(29,68,48,35,62,42)(30,43,63,36,37,69)(73,125,117,79,131,111)(74,112,132,80,118,126)(75,127,119,81,121,113)(76,114,122,82,120,128)(77,129,109,83,123,115)(78,116,124,84,110,130)(85,101,139,91,107,133)(86,134,108,92,140,102)(87,103,141,93,97,135)(88,136,98,94,142,104)(89,105,143,95,99,137)(90,138,100,96,144,106), (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), (1,30,115,105)(2,29,116,104)(3,28,117,103)(4,27,118,102)(5,26,119,101)(6,25,120,100)(7,36,109,99)(8,35,110,98)(9,34,111,97)(10,33,112,108)(11,32,113,107)(12,31,114,106)(13,45,121,133)(14,44,122,144)(15,43,123,143)(16,42,124,142)(17,41,125,141)(18,40,126,140)(19,39,127,139)(20,38,128,138)(21,37,129,137)(22,48,130,136)(23,47,131,135)(24,46,132,134)(49,63,83,95)(50,62,84,94)(51,61,73,93)(52,72,74,92)(53,71,75,91)(54,70,76,90)(55,69,77,89)(56,68,78,88)(57,67,79,87)(58,66,80,86)(59,65,81,85)(60,64,82,96) );
G=PermutationGroup([[(1,121,57,115,13,79),(2,122,58,116,14,80),(3,123,59,117,15,81),(4,124,60,118,16,82),(5,125,49,119,17,83),(6,126,50,120,18,84),(7,127,51,109,19,73),(8,128,52,110,20,74),(9,129,53,111,21,75),(10,130,54,112,22,76),(11,131,55,113,23,77),(12,132,56,114,24,78),(25,94,40,100,62,140),(26,95,41,101,63,141),(27,96,42,102,64,142),(28,85,43,103,65,143),(29,86,44,104,66,144),(30,87,45,105,67,133),(31,88,46,106,68,134),(32,89,47,107,69,135),(33,90,48,108,70,136),(34,91,37,97,71,137),(35,92,38,98,72,138),(36,93,39,99,61,139)], [(1,55,21,7,49,15),(2,16,50,8,22,56),(3,57,23,9,51,17),(4,18,52,10,24,58),(5,59,13,11,53,19),(6,20,54,12,14,60),(25,64,44,31,70,38),(26,39,71,32,45,65),(27,66,46,33,72,40),(28,41,61,34,47,67),(29,68,48,35,62,42),(30,43,63,36,37,69),(73,125,117,79,131,111),(74,112,132,80,118,126),(75,127,119,81,121,113),(76,114,122,82,120,128),(77,129,109,83,123,115),(78,116,124,84,110,130),(85,101,139,91,107,133),(86,134,108,92,140,102),(87,103,141,93,97,135),(88,136,98,94,142,104),(89,105,143,95,99,137),(90,138,100,96,144,106)], [(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)], [(1,30,115,105),(2,29,116,104),(3,28,117,103),(4,27,118,102),(5,26,119,101),(6,25,120,100),(7,36,109,99),(8,35,110,98),(9,34,111,97),(10,33,112,108),(11,32,113,107),(12,31,114,106),(13,45,121,133),(14,44,122,144),(15,43,123,143),(16,42,124,142),(17,41,125,141),(18,40,126,140),(19,39,127,139),(20,38,128,138),(21,37,129,137),(22,48,130,136),(23,47,131,135),(24,46,132,134),(49,63,83,95),(50,62,84,94),(51,61,73,93),(52,72,74,92),(53,71,75,91),(54,70,76,90),(55,69,77,89),(56,68,78,88),(57,67,79,87),(58,66,80,86),(59,65,81,85),(60,64,82,96)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6O | 6P | ··· | 6AA | 12A | ··· | 12P | 12Q | 12R | 12S | 12T |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 18 | 18 | 54 | 54 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 18 | 18 | 18 | 18 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | - | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 | S32 | S3×Dic3 | C3⋊D12 | C32⋊2Q8 |
| kernel | C62.80D6 | Dic3×C3×C6 | C6×C3⋊Dic3 | C2×C33⋊5C4 | C32×Dic3 | C6×Dic3 | C2×C3⋊Dic3 | C32×C6 | C32×C6 | C3×Dic3 | C62 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 8 | 5 | 10 | 2 | 8 | 2 | 4 | 4 | 4 | 4 |
Matrix representation of C62.80D6 ►in GL6(𝔽13)
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [10,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[11,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,11,0,0,0,0,7,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C62.80D6 in GAP, Magma, Sage, TeX
C_6^2._{80}D_6 % in TeX
G:=Group("C6^2.80D6"); // GroupNames label
G:=SmallGroup(432,452);
// by ID
G=gap.SmallGroup(432,452);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^3,d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^5>;
// generators/relations