metabelian, supersoluble, monomial
Aliases: C62.75D4, (C3×D4).44D6, (C3×Q8).68D6, (C2×C12).163D6, (C3×C12).155D4, C3⋊5(Q8.14D6), C32⋊7Q16⋊10C2, C32⋊9SD16⋊10C2, C12.58D6⋊16C2, C12.120(C3⋊D4), C12.105(C22×S3), (C6×C12).155C22, (C3×C12).109C23, C4.25(C32⋊7D4), C32⋊25(C8.C22), C32⋊4C8.18C22, (D4×C32).29C22, C22.6(C32⋊7D4), (Q8×C32).30C22, C32⋊4Q8.33C22, D4.9(C2×C3⋊S3), Q8.14(C2×C3⋊S3), C4○D4.4(C3⋊S3), (C3×C4○D4).19S3, (C3×C6).295(C2×D4), C6.136(C2×C3⋊D4), C4.19(C22×C3⋊S3), (C2×C6).28(C3⋊D4), (C32×C4○D4).6C2, (C2×C32⋊4Q8)⋊17C2, C2.25(C2×C32⋊7D4), (C2×C4).22(C2×C3⋊S3), SmallGroup(288,808)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.75D4
G = < a,b,c,d | a6=b6=1, c4=d2=b3, ab=ba, cac-1=a-1b3, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=c3 >
Subgroups: 556 in 180 conjugacy classes, 65 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×C6, C3×C6, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8.C22, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C4.Dic3, D4.S3, C3⋊Q16, C2×Dic6, C3×C4○D4, C32⋊4C8, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8.14D6, C12.58D6, C32⋊9SD16, C32⋊7Q16, C2×C32⋊4Q8, C32×C4○D4, C62.75D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C8.C22, C2×C3⋊S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, Q8.14D6, C2×C32⋊7D4, C62.75D4
►On 144 points
Generators in S144
(1 87 141 99 118 121)(2 126 119 104 142 84)(3 81 143 101 120 123)(4 128 113 98 144 86)(5 83 137 103 114 125)(6 122 115 100 138 88)(7 85 139 97 116 127)(8 124 117 102 140 82)(9 90 48 132 58 20)(10 17 59 129 41 95)(11 92 42 134 60 22)(12 19 61 131 43 89)(13 94 44 136 62 24)(14 21 63 133 45 91)(15 96 46 130 64 18)(16 23 57 135 47 93)(25 78 53 110 35 67)(26 72 36 107 54 75)(27 80 55 112 37 69)(28 66 38 109 56 77)(29 74 49 106 39 71)(30 68 40 111 50 79)(31 76 51 108 33 65)(32 70 34 105 52 73)
(1 112 48 5 108 44)(2 45 109 6 41 105)(3 106 42 7 110 46)(4 47 111 8 43 107)(9 114 76 13 118 80)(10 73 119 14 77 115)(11 116 78 15 120 74)(12 75 113 16 79 117)(17 32 104 21 28 100)(18 101 29 22 97 25)(19 26 98 23 30 102)(20 103 31 24 99 27)(33 136 87 37 132 83)(34 84 133 38 88 129)(35 130 81 39 134 85)(36 86 135 40 82 131)(49 92 127 53 96 123)(50 124 89 54 128 93)(51 94 121 55 90 125)(52 126 91 56 122 95)(57 68 140 61 72 144)(58 137 65 62 141 69)(59 70 142 63 66 138)(60 139 67 64 143 71)
(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 4 5 8)(2 7 6 3)(9 12 13 16)(10 15 14 11)(17 18 21 22)(19 24 23 20)(25 28 29 32)(26 31 30 27)(33 40 37 36)(34 35 38 39)(41 46 45 42)(43 44 47 48)(49 52 53 56)(50 55 54 51)(57 58 61 62)(59 64 63 60)(65 68 69 72)(66 71 70 67)(73 78 77 74)(75 76 79 80)(81 84 85 88)(82 87 86 83)(89 94 93 90)(91 92 95 96)(97 100 101 104)(98 103 102 99)(105 110 109 106)(107 108 111 112)(113 114 117 118)(115 120 119 116)(121 128 125 124)(122 123 126 127)(129 130 133 134)(131 136 135 132)(137 140 141 144)(138 143 142 139)
G:=sub<Sym(144)| (1,87,141,99,118,121)(2,126,119,104,142,84)(3,81,143,101,120,123)(4,128,113,98,144,86)(5,83,137,103,114,125)(6,122,115,100,138,88)(7,85,139,97,116,127)(8,124,117,102,140,82)(9,90,48,132,58,20)(10,17,59,129,41,95)(11,92,42,134,60,22)(12,19,61,131,43,89)(13,94,44,136,62,24)(14,21,63,133,45,91)(15,96,46,130,64,18)(16,23,57,135,47,93)(25,78,53,110,35,67)(26,72,36,107,54,75)(27,80,55,112,37,69)(28,66,38,109,56,77)(29,74,49,106,39,71)(30,68,40,111,50,79)(31,76,51,108,33,65)(32,70,34,105,52,73), (1,112,48,5,108,44)(2,45,109,6,41,105)(3,106,42,7,110,46)(4,47,111,8,43,107)(9,114,76,13,118,80)(10,73,119,14,77,115)(11,116,78,15,120,74)(12,75,113,16,79,117)(17,32,104,21,28,100)(18,101,29,22,97,25)(19,26,98,23,30,102)(20,103,31,24,99,27)(33,136,87,37,132,83)(34,84,133,38,88,129)(35,130,81,39,134,85)(36,86,135,40,82,131)(49,92,127,53,96,123)(50,124,89,54,128,93)(51,94,121,55,90,125)(52,126,91,56,122,95)(57,68,140,61,72,144)(58,137,65,62,141,69)(59,70,142,63,66,138)(60,139,67,64,143,71), (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), (1,4,5,8)(2,7,6,3)(9,12,13,16)(10,15,14,11)(17,18,21,22)(19,24,23,20)(25,28,29,32)(26,31,30,27)(33,40,37,36)(34,35,38,39)(41,46,45,42)(43,44,47,48)(49,52,53,56)(50,55,54,51)(57,58,61,62)(59,64,63,60)(65,68,69,72)(66,71,70,67)(73,78,77,74)(75,76,79,80)(81,84,85,88)(82,87,86,83)(89,94,93,90)(91,92,95,96)(97,100,101,104)(98,103,102,99)(105,110,109,106)(107,108,111,112)(113,114,117,118)(115,120,119,116)(121,128,125,124)(122,123,126,127)(129,130,133,134)(131,136,135,132)(137,140,141,144)(138,143,142,139)>;
G:=Group( (1,87,141,99,118,121)(2,126,119,104,142,84)(3,81,143,101,120,123)(4,128,113,98,144,86)(5,83,137,103,114,125)(6,122,115,100,138,88)(7,85,139,97,116,127)(8,124,117,102,140,82)(9,90,48,132,58,20)(10,17,59,129,41,95)(11,92,42,134,60,22)(12,19,61,131,43,89)(13,94,44,136,62,24)(14,21,63,133,45,91)(15,96,46,130,64,18)(16,23,57,135,47,93)(25,78,53,110,35,67)(26,72,36,107,54,75)(27,80,55,112,37,69)(28,66,38,109,56,77)(29,74,49,106,39,71)(30,68,40,111,50,79)(31,76,51,108,33,65)(32,70,34,105,52,73), (1,112,48,5,108,44)(2,45,109,6,41,105)(3,106,42,7,110,46)(4,47,111,8,43,107)(9,114,76,13,118,80)(10,73,119,14,77,115)(11,116,78,15,120,74)(12,75,113,16,79,117)(17,32,104,21,28,100)(18,101,29,22,97,25)(19,26,98,23,30,102)(20,103,31,24,99,27)(33,136,87,37,132,83)(34,84,133,38,88,129)(35,130,81,39,134,85)(36,86,135,40,82,131)(49,92,127,53,96,123)(50,124,89,54,128,93)(51,94,121,55,90,125)(52,126,91,56,122,95)(57,68,140,61,72,144)(58,137,65,62,141,69)(59,70,142,63,66,138)(60,139,67,64,143,71), (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), (1,4,5,8)(2,7,6,3)(9,12,13,16)(10,15,14,11)(17,18,21,22)(19,24,23,20)(25,28,29,32)(26,31,30,27)(33,40,37,36)(34,35,38,39)(41,46,45,42)(43,44,47,48)(49,52,53,56)(50,55,54,51)(57,58,61,62)(59,64,63,60)(65,68,69,72)(66,71,70,67)(73,78,77,74)(75,76,79,80)(81,84,85,88)(82,87,86,83)(89,94,93,90)(91,92,95,96)(97,100,101,104)(98,103,102,99)(105,110,109,106)(107,108,111,112)(113,114,117,118)(115,120,119,116)(121,128,125,124)(122,123,126,127)(129,130,133,134)(131,136,135,132)(137,140,141,144)(138,143,142,139) );
G=PermutationGroup([[(1,87,141,99,118,121),(2,126,119,104,142,84),(3,81,143,101,120,123),(4,128,113,98,144,86),(5,83,137,103,114,125),(6,122,115,100,138,88),(7,85,139,97,116,127),(8,124,117,102,140,82),(9,90,48,132,58,20),(10,17,59,129,41,95),(11,92,42,134,60,22),(12,19,61,131,43,89),(13,94,44,136,62,24),(14,21,63,133,45,91),(15,96,46,130,64,18),(16,23,57,135,47,93),(25,78,53,110,35,67),(26,72,36,107,54,75),(27,80,55,112,37,69),(28,66,38,109,56,77),(29,74,49,106,39,71),(30,68,40,111,50,79),(31,76,51,108,33,65),(32,70,34,105,52,73)], [(1,112,48,5,108,44),(2,45,109,6,41,105),(3,106,42,7,110,46),(4,47,111,8,43,107),(9,114,76,13,118,80),(10,73,119,14,77,115),(11,116,78,15,120,74),(12,75,113,16,79,117),(17,32,104,21,28,100),(18,101,29,22,97,25),(19,26,98,23,30,102),(20,103,31,24,99,27),(33,136,87,37,132,83),(34,84,133,38,88,129),(35,130,81,39,134,85),(36,86,135,40,82,131),(49,92,127,53,96,123),(50,124,89,54,128,93),(51,94,121,55,90,125),(52,126,91,56,122,95),(57,68,140,61,72,144),(58,137,65,62,141,69),(59,70,142,63,66,138),(60,139,67,64,143,71)], [(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)], [(1,4,5,8),(2,7,6,3),(9,12,13,16),(10,15,14,11),(17,18,21,22),(19,24,23,20),(25,28,29,32),(26,31,30,27),(33,40,37,36),(34,35,38,39),(41,46,45,42),(43,44,47,48),(49,52,53,56),(50,55,54,51),(57,58,61,62),(59,64,63,60),(65,68,69,72),(66,71,70,67),(73,78,77,74),(75,76,79,80),(81,84,85,88),(82,87,86,83),(89,94,93,90),(91,92,95,96),(97,100,101,104),(98,103,102,99),(105,110,109,106),(107,108,111,112),(113,114,117,118),(115,120,119,116),(121,128,125,124),(122,123,126,127),(129,130,133,134),(131,136,135,132),(137,140,141,144),(138,143,142,139)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | ··· | 6P | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 36 | 36 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C8.C22 | Q8.14D6 |
| kernel | C62.75D4 | C12.58D6 | C32⋊9SD16 | C32⋊7Q16 | C2×C32⋊4Q8 | C32×C4○D4 | C3×C4○D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×Q8 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 1 | 8 |
Matrix representation of C62.75D4 ►in GL8(𝔽73)
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 51 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 22 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 36 | 45 | 45 |
| 0 | 0 | 0 | 0 | 37 | 37 | 28 | 45 |
| 0 | 0 | 0 | 0 | 28 | 28 | 36 | 37 |
| 0 | 0 | 0 | 0 | 45 | 28 | 36 | 36 |
| 51 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 22 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 36 | 45 | 45 |
| 0 | 0 | 0 | 0 | 36 | 36 | 45 | 28 |
| 0 | 0 | 0 | 0 | 45 | 45 | 37 | 36 |
| 0 | 0 | 0 | 0 | 45 | 28 | 36 | 36 |
G:=sub<GL(8,GF(73))| [1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[51,32,0,0,0,0,0,0,10,22,0,0,0,0,0,0,0,0,42,28,0,0,0,0,0,0,70,31,0,0,0,0,0,0,0,0,37,37,28,45,0,0,0,0,36,37,28,28,0,0,0,0,45,28,36,36,0,0,0,0,45,45,37,36],[51,32,0,0,0,0,0,0,10,22,0,0,0,0,0,0,0,0,42,28,0,0,0,0,0,0,70,31,0,0,0,0,0,0,0,0,37,36,45,45,0,0,0,0,36,36,45,28,0,0,0,0,45,45,37,36,0,0,0,0,45,28,36,36] >;
C62.75D4 in GAP, Magma, Sage, TeX
C_6^2._{75}D_4 % in TeX
G:=Group("C6^2.75D4"); // GroupNames label
G:=SmallGroup(288,808);
// by ID
G=gap.SmallGroup(288,808);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,675,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^4=d^2=b^3,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations