metabelian, supersoluble, monomial
Aliases: C62.74D4, (C3×D4).43D6, (C3×Q8).67D6, (C3×C12).175D4, (C2×C12).162D6, C32⋊25(C4○D8), C32⋊7D8⋊11C2, C12.59D6⋊6C2, C3⋊7(Q8.13D6), C32⋊7Q16⋊11C2, C32⋊11SD16⋊11C2, C32⋊9SD16⋊11C2, C12.135(C3⋊D4), (C6×C12).154C22, (C3×C12).108C23, C12.104(C22×S3), C4.32(C32⋊7D4), C12⋊S3.32C22, C32⋊4C8.31C22, (D4×C32).28C22, C22.1(C32⋊7D4), (Q8×C32).29C22, C32⋊4Q8.32C22, (C3×C4○D4)⋊6S3, D4.8(C2×C3⋊S3), C4○D4⋊4(C3⋊S3), Q8.13(C2×C3⋊S3), (C3×C6).294(C2×D4), (C32×C4○D4)⋊4C2, C6.135(C2×C3⋊D4), C4.18(C22×C3⋊S3), (C2×C32⋊4C8)⋊13C2, (C2×C6).27(C3⋊D4), C2.24(C2×C32⋊7D4), (C2×C4).59(C2×C3⋊S3), SmallGroup(288,807)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.74D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a-1, dad=a-1b3, cbc-1=dbd=b-1, dcd=b3c3 >
Subgroups: 644 in 186 conjugacy classes, 65 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C62, C62, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, C32⋊4C8, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8.13D6, C2×C32⋊4C8, C32⋊7D8, C32⋊9SD16, C32⋊11SD16, C32⋊7Q16, C12.59D6, C32×C4○D4, C62.74D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C4○D8, C2×C3⋊S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, Q8.13D6, C2×C32⋊7D4, C62.74D4
►On 144 points
Generators in S144
(1 82 94 45 68 57)(2 58 69 46 95 83)(3 84 96 47 70 59)(4 60 71 48 89 85)(5 86 90 41 72 61)(6 62 65 42 91 87)(7 88 92 43 66 63)(8 64 67 44 93 81)(9 117 49 106 21 135)(10 136 22 107 50 118)(11 119 51 108 23 129)(12 130 24 109 52 120)(13 113 53 110 17 131)(14 132 18 111 54 114)(15 115 55 112 19 133)(16 134 20 105 56 116)(25 74 139 99 36 123)(26 124 37 100 140 75)(27 76 141 101 38 125)(28 126 39 102 142 77)(29 78 143 103 40 127)(30 128 33 104 144 79)(31 80 137 97 34 121)(32 122 35 98 138 73)
(1 103 118 5 99 114)(2 115 100 6 119 104)(3 97 120 7 101 116)(4 117 102 8 113 98)(9 39 81 13 35 85)(10 86 36 14 82 40)(11 33 83 15 37 87)(12 88 38 16 84 34)(17 32 48 21 28 44)(18 45 29 22 41 25)(19 26 42 23 30 46)(20 47 31 24 43 27)(49 142 64 53 138 60)(50 61 139 54 57 143)(51 144 58 55 140 62)(52 63 141 56 59 137)(65 108 79 69 112 75)(66 76 105 70 80 109)(67 110 73 71 106 77)(68 78 107 72 74 111)(89 135 126 93 131 122)(90 123 132 94 127 136)(91 129 128 95 133 124)(92 125 134 96 121 130)
(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)
(2 8)(3 7)(4 6)(9 144)(10 143)(11 142)(12 141)(13 140)(14 139)(15 138)(16 137)(17 26)(18 25)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(33 49)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(41 45)(42 44)(46 48)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 88)(64 87)(65 89)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 129)(74 136)(75 135)(76 134)(77 133)(78 132)(79 131)(80 130)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 125)(106 124)(107 123)(108 122)(109 121)(110 128)(111 127)(112 126)
G:=sub<Sym(144)| (1,82,94,45,68,57)(2,58,69,46,95,83)(3,84,96,47,70,59)(4,60,71,48,89,85)(5,86,90,41,72,61)(6,62,65,42,91,87)(7,88,92,43,66,63)(8,64,67,44,93,81)(9,117,49,106,21,135)(10,136,22,107,50,118)(11,119,51,108,23,129)(12,130,24,109,52,120)(13,113,53,110,17,131)(14,132,18,111,54,114)(15,115,55,112,19,133)(16,134,20,105,56,116)(25,74,139,99,36,123)(26,124,37,100,140,75)(27,76,141,101,38,125)(28,126,39,102,142,77)(29,78,143,103,40,127)(30,128,33,104,144,79)(31,80,137,97,34,121)(32,122,35,98,138,73), (1,103,118,5,99,114)(2,115,100,6,119,104)(3,97,120,7,101,116)(4,117,102,8,113,98)(9,39,81,13,35,85)(10,86,36,14,82,40)(11,33,83,15,37,87)(12,88,38,16,84,34)(17,32,48,21,28,44)(18,45,29,22,41,25)(19,26,42,23,30,46)(20,47,31,24,43,27)(49,142,64,53,138,60)(50,61,139,54,57,143)(51,144,58,55,140,62)(52,63,141,56,59,137)(65,108,79,69,112,75)(66,76,105,70,80,109)(67,110,73,71,106,77)(68,78,107,72,74,111)(89,135,126,93,131,122)(90,123,132,94,127,136)(91,129,128,95,133,124)(92,125,134,96,121,130), (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), (2,8)(3,7)(4,6)(9,144)(10,143)(11,142)(12,141)(13,140)(14,139)(15,138)(16,137)(17,26)(18,25)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(33,49)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,45)(42,44)(46,48)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,88)(64,87)(65,89)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,129)(74,136)(75,135)(76,134)(77,133)(78,132)(79,131)(80,130)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,125)(106,124)(107,123)(108,122)(109,121)(110,128)(111,127)(112,126)>;
G:=Group( (1,82,94,45,68,57)(2,58,69,46,95,83)(3,84,96,47,70,59)(4,60,71,48,89,85)(5,86,90,41,72,61)(6,62,65,42,91,87)(7,88,92,43,66,63)(8,64,67,44,93,81)(9,117,49,106,21,135)(10,136,22,107,50,118)(11,119,51,108,23,129)(12,130,24,109,52,120)(13,113,53,110,17,131)(14,132,18,111,54,114)(15,115,55,112,19,133)(16,134,20,105,56,116)(25,74,139,99,36,123)(26,124,37,100,140,75)(27,76,141,101,38,125)(28,126,39,102,142,77)(29,78,143,103,40,127)(30,128,33,104,144,79)(31,80,137,97,34,121)(32,122,35,98,138,73), (1,103,118,5,99,114)(2,115,100,6,119,104)(3,97,120,7,101,116)(4,117,102,8,113,98)(9,39,81,13,35,85)(10,86,36,14,82,40)(11,33,83,15,37,87)(12,88,38,16,84,34)(17,32,48,21,28,44)(18,45,29,22,41,25)(19,26,42,23,30,46)(20,47,31,24,43,27)(49,142,64,53,138,60)(50,61,139,54,57,143)(51,144,58,55,140,62)(52,63,141,56,59,137)(65,108,79,69,112,75)(66,76,105,70,80,109)(67,110,73,71,106,77)(68,78,107,72,74,111)(89,135,126,93,131,122)(90,123,132,94,127,136)(91,129,128,95,133,124)(92,125,134,96,121,130), (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), (2,8)(3,7)(4,6)(9,144)(10,143)(11,142)(12,141)(13,140)(14,139)(15,138)(16,137)(17,26)(18,25)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(33,49)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,45)(42,44)(46,48)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,88)(64,87)(65,89)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,129)(74,136)(75,135)(76,134)(77,133)(78,132)(79,131)(80,130)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,125)(106,124)(107,123)(108,122)(109,121)(110,128)(111,127)(112,126) );
G=PermutationGroup([[(1,82,94,45,68,57),(2,58,69,46,95,83),(3,84,96,47,70,59),(4,60,71,48,89,85),(5,86,90,41,72,61),(6,62,65,42,91,87),(7,88,92,43,66,63),(8,64,67,44,93,81),(9,117,49,106,21,135),(10,136,22,107,50,118),(11,119,51,108,23,129),(12,130,24,109,52,120),(13,113,53,110,17,131),(14,132,18,111,54,114),(15,115,55,112,19,133),(16,134,20,105,56,116),(25,74,139,99,36,123),(26,124,37,100,140,75),(27,76,141,101,38,125),(28,126,39,102,142,77),(29,78,143,103,40,127),(30,128,33,104,144,79),(31,80,137,97,34,121),(32,122,35,98,138,73)], [(1,103,118,5,99,114),(2,115,100,6,119,104),(3,97,120,7,101,116),(4,117,102,8,113,98),(9,39,81,13,35,85),(10,86,36,14,82,40),(11,33,83,15,37,87),(12,88,38,16,84,34),(17,32,48,21,28,44),(18,45,29,22,41,25),(19,26,42,23,30,46),(20,47,31,24,43,27),(49,142,64,53,138,60),(50,61,139,54,57,143),(51,144,58,55,140,62),(52,63,141,56,59,137),(65,108,79,69,112,75),(66,76,105,70,80,109),(67,110,73,71,106,77),(68,78,107,72,74,111),(89,135,126,93,131,122),(90,123,132,94,127,136),(91,129,128,95,133,124),(92,125,134,96,121,130)], [(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)], [(2,8),(3,7),(4,6),(9,144),(10,143),(11,142),(12,141),(13,140),(14,139),(15,138),(16,137),(17,26),(18,25),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(33,49),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50),(41,45),(42,44),(46,48),(57,86),(58,85),(59,84),(60,83),(61,82),(62,81),(63,88),(64,87),(65,89),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90),(73,129),(74,136),(75,135),(76,134),(77,133),(78,132),(79,131),(80,130),(97,120),(98,119),(99,118),(100,117),(101,116),(102,115),(103,114),(104,113),(105,125),(106,124),(107,123),(108,122),(109,121),(110,128),(111,127),(112,126)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | ··· | 6P | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 4 | 36 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 36 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C4○D8 | Q8.13D6 |
| kernel | C62.74D4 | C2×C32⋊4C8 | C32⋊7D8 | C32⋊9SD16 | C32⋊11SD16 | C32⋊7Q16 | C12.59D6 | C32×C4○D4 | C3×C4○D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×Q8 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 4 | 8 |
Matrix representation of C62.74D4 ►in GL6(𝔽73)
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 19 |
| 0 | 0 | 0 | 0 | 27 | 27 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 60 | 43 | 0 | 0 | 0 | 0 |
| 30 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 60 | 0 | 0 |
| 0 | 0 | 30 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 32 |
| 0 | 0 | 0 | 0 | 57 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,46,27,0,0,0,0,19,27],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[60,30,0,0,0,0,43,13,0,0,0,0,0,0,30,30,0,0,0,0,60,43,0,0,0,0,0,0,32,57,0,0,0,0,32,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
C62.74D4 in GAP, Magma, Sage, TeX
C_6^2._{74}D_4 % in TeX
G:=Group("C6^2.74D4"); // GroupNames label
G:=SmallGroup(288,807);
// by ID
G=gap.SmallGroup(288,807);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,675,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^4=b^3,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^3>;
// generators/relations