metabelian, supersoluble, monomial
Aliases: C62⋊19D4, C62.251C23, (C2×C6)⋊8D12, (C3×C12)⋊22D4, C6.64(C2×D12), C3⋊5(C12⋊7D4), C12⋊12(C3⋊D4), (C22×C12)⋊10S3, (C2×C12).389D6, C4⋊3(C32⋊7D4), C6.11D12⋊3C2, C32⋊21(C4⋊D4), (C22×C6).160D6, C22⋊2(C12⋊S3), C6.109(C4○D12), C12⋊Dic3⋊12C2, (C6×C12).291C22, (C2×C62).112C22, C2.19(C12.59D6), (C2×C6×C12)⋊8C2, (C2×C12⋊S3)⋊8C2, (C22×C4)⋊6(C3⋊S3), (C3×C6).277(C2×D4), C6.118(C2×C3⋊D4), (C2×C32⋊7D4)⋊9C2, C23.29(C2×C3⋊S3), C2.17(C2×C12⋊S3), C2.7(C2×C32⋊7D4), (C3×C6).124(C4○D4), (C2×C6).268(C22×S3), C22.56(C22×C3⋊S3), (C22×C3⋊S3).45C22, (C2×C3⋊Dic3).91C22, (C2×C4).69(C2×C3⋊S3), SmallGroup(288,787)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62⋊19D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, ac=ca, dad=a-1b3, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 1196 in 282 conjugacy classes, 89 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4⋊D4, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C62, C62, C62, C4⋊Dic3, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C12⋊S3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, C6×C12, C22×C3⋊S3, C2×C62, C12⋊7D4, C12⋊Dic3, C6.11D12, C2×C12⋊S3, C2×C32⋊7D4, C2×C6×C12, C62⋊19D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, D12, C3⋊D4, C22×S3, C4⋊D4, C2×C3⋊S3, C2×D12, C4○D12, C2×C3⋊D4, C12⋊S3, C32⋊7D4, C22×C3⋊S3, C12⋊7D4, C2×C12⋊S3, C12.59D6, C2×C32⋊7D4, C62⋊19D4
►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 75 35 51 25 112)(2 76 36 52 26 113)(3 77 31 53 27 114)(4 78 32 54 28 109)(5 73 33 49 29 110)(6 74 34 50 30 111)(7 106 122 102 47 90)(8 107 123 97 48 85)(9 108 124 98 43 86)(10 103 125 99 44 87)(11 104 126 100 45 88)(12 105 121 101 46 89)(13 66 116 57 24 68)(14 61 117 58 19 69)(15 62 118 59 20 70)(16 63 119 60 21 71)(17 64 120 55 22 72)(18 65 115 56 23 67)(37 130 91 141 80 133)(38 131 92 142 81 134)(39 132 93 143 82 135)(40 127 94 144 83 136)(41 128 95 139 84 137)(42 129 96 140 79 138)
(1 128 56 85)(2 129 57 86)(3 130 58 87)(4 131 59 88)(5 132 60 89)(6 127 55 90)(7 74 94 22)(8 75 95 23)(9 76 96 24)(10 77 91 19)(11 78 92 20)(12 73 93 21)(13 124 52 79)(14 125 53 80)(15 126 54 81)(16 121 49 82)(17 122 50 83)(18 123 51 84)(25 137 65 97)(26 138 66 98)(27 133 61 99)(28 134 62 100)(29 135 63 101)(30 136 64 102)(31 141 69 103)(32 142 70 104)(33 143 71 105)(34 144 72 106)(35 139 67 107)(36 140 68 108)(37 117 44 114)(38 118 45 109)(39 119 46 110)(40 120 47 111)(41 115 48 112)(42 116 43 113)
(1 85)(2 122)(3 89)(4 126)(5 87)(6 124)(7 36)(8 112)(9 34)(10 110)(11 32)(12 114)(13 127)(14 82)(15 131)(16 80)(17 129)(18 84)(19 39)(20 142)(21 37)(22 140)(23 41)(24 144)(25 107)(26 47)(27 105)(28 45)(29 103)(30 43)(31 101)(33 99)(35 97)(38 62)(40 66)(42 64)(44 73)(46 77)(48 75)(49 125)(50 86)(51 123)(52 90)(53 121)(54 88)(55 79)(56 128)(57 83)(58 132)(59 81)(60 130)(61 143)(63 141)(65 139)(67 137)(68 94)(69 135)(70 92)(71 133)(72 96)(74 108)(76 106)(78 104)(91 119)(93 117)(95 115)(98 111)(100 109)(102 113)(116 136)(118 134)(120 138)
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), (1,75,35,51,25,112)(2,76,36,52,26,113)(3,77,31,53,27,114)(4,78,32,54,28,109)(5,73,33,49,29,110)(6,74,34,50,30,111)(7,106,122,102,47,90)(8,107,123,97,48,85)(9,108,124,98,43,86)(10,103,125,99,44,87)(11,104,126,100,45,88)(12,105,121,101,46,89)(13,66,116,57,24,68)(14,61,117,58,19,69)(15,62,118,59,20,70)(16,63,119,60,21,71)(17,64,120,55,22,72)(18,65,115,56,23,67)(37,130,91,141,80,133)(38,131,92,142,81,134)(39,132,93,143,82,135)(40,127,94,144,83,136)(41,128,95,139,84,137)(42,129,96,140,79,138), (1,128,56,85)(2,129,57,86)(3,130,58,87)(4,131,59,88)(5,132,60,89)(6,127,55,90)(7,74,94,22)(8,75,95,23)(9,76,96,24)(10,77,91,19)(11,78,92,20)(12,73,93,21)(13,124,52,79)(14,125,53,80)(15,126,54,81)(16,121,49,82)(17,122,50,83)(18,123,51,84)(25,137,65,97)(26,138,66,98)(27,133,61,99)(28,134,62,100)(29,135,63,101)(30,136,64,102)(31,141,69,103)(32,142,70,104)(33,143,71,105)(34,144,72,106)(35,139,67,107)(36,140,68,108)(37,117,44,114)(38,118,45,109)(39,119,46,110)(40,120,47,111)(41,115,48,112)(42,116,43,113), (1,85)(2,122)(3,89)(4,126)(5,87)(6,124)(7,36)(8,112)(9,34)(10,110)(11,32)(12,114)(13,127)(14,82)(15,131)(16,80)(17,129)(18,84)(19,39)(20,142)(21,37)(22,140)(23,41)(24,144)(25,107)(26,47)(27,105)(28,45)(29,103)(30,43)(31,101)(33,99)(35,97)(38,62)(40,66)(42,64)(44,73)(46,77)(48,75)(49,125)(50,86)(51,123)(52,90)(53,121)(54,88)(55,79)(56,128)(57,83)(58,132)(59,81)(60,130)(61,143)(63,141)(65,139)(67,137)(68,94)(69,135)(70,92)(71,133)(72,96)(74,108)(76,106)(78,104)(91,119)(93,117)(95,115)(98,111)(100,109)(102,113)(116,136)(118,134)(120,138)>;
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), (1,75,35,51,25,112)(2,76,36,52,26,113)(3,77,31,53,27,114)(4,78,32,54,28,109)(5,73,33,49,29,110)(6,74,34,50,30,111)(7,106,122,102,47,90)(8,107,123,97,48,85)(9,108,124,98,43,86)(10,103,125,99,44,87)(11,104,126,100,45,88)(12,105,121,101,46,89)(13,66,116,57,24,68)(14,61,117,58,19,69)(15,62,118,59,20,70)(16,63,119,60,21,71)(17,64,120,55,22,72)(18,65,115,56,23,67)(37,130,91,141,80,133)(38,131,92,142,81,134)(39,132,93,143,82,135)(40,127,94,144,83,136)(41,128,95,139,84,137)(42,129,96,140,79,138), (1,128,56,85)(2,129,57,86)(3,130,58,87)(4,131,59,88)(5,132,60,89)(6,127,55,90)(7,74,94,22)(8,75,95,23)(9,76,96,24)(10,77,91,19)(11,78,92,20)(12,73,93,21)(13,124,52,79)(14,125,53,80)(15,126,54,81)(16,121,49,82)(17,122,50,83)(18,123,51,84)(25,137,65,97)(26,138,66,98)(27,133,61,99)(28,134,62,100)(29,135,63,101)(30,136,64,102)(31,141,69,103)(32,142,70,104)(33,143,71,105)(34,144,72,106)(35,139,67,107)(36,140,68,108)(37,117,44,114)(38,118,45,109)(39,119,46,110)(40,120,47,111)(41,115,48,112)(42,116,43,113), (1,85)(2,122)(3,89)(4,126)(5,87)(6,124)(7,36)(8,112)(9,34)(10,110)(11,32)(12,114)(13,127)(14,82)(15,131)(16,80)(17,129)(18,84)(19,39)(20,142)(21,37)(22,140)(23,41)(24,144)(25,107)(26,47)(27,105)(28,45)(29,103)(30,43)(31,101)(33,99)(35,97)(38,62)(40,66)(42,64)(44,73)(46,77)(48,75)(49,125)(50,86)(51,123)(52,90)(53,121)(54,88)(55,79)(56,128)(57,83)(58,132)(59,81)(60,130)(61,143)(63,141)(65,139)(67,137)(68,94)(69,135)(70,92)(71,133)(72,96)(74,108)(76,106)(78,104)(91,119)(93,117)(95,115)(98,111)(100,109)(102,113)(116,136)(118,134)(120,138) );
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)], [(1,75,35,51,25,112),(2,76,36,52,26,113),(3,77,31,53,27,114),(4,78,32,54,28,109),(5,73,33,49,29,110),(6,74,34,50,30,111),(7,106,122,102,47,90),(8,107,123,97,48,85),(9,108,124,98,43,86),(10,103,125,99,44,87),(11,104,126,100,45,88),(12,105,121,101,46,89),(13,66,116,57,24,68),(14,61,117,58,19,69),(15,62,118,59,20,70),(16,63,119,60,21,71),(17,64,120,55,22,72),(18,65,115,56,23,67),(37,130,91,141,80,133),(38,131,92,142,81,134),(39,132,93,143,82,135),(40,127,94,144,83,136),(41,128,95,139,84,137),(42,129,96,140,79,138)], [(1,128,56,85),(2,129,57,86),(3,130,58,87),(4,131,59,88),(5,132,60,89),(6,127,55,90),(7,74,94,22),(8,75,95,23),(9,76,96,24),(10,77,91,19),(11,78,92,20),(12,73,93,21),(13,124,52,79),(14,125,53,80),(15,126,54,81),(16,121,49,82),(17,122,50,83),(18,123,51,84),(25,137,65,97),(26,138,66,98),(27,133,61,99),(28,134,62,100),(29,135,63,101),(30,136,64,102),(31,141,69,103),(32,142,70,104),(33,143,71,105),(34,144,72,106),(35,139,67,107),(36,140,68,108),(37,117,44,114),(38,118,45,109),(39,119,46,110),(40,120,47,111),(41,115,48,112),(42,116,43,113)], [(1,85),(2,122),(3,89),(4,126),(5,87),(6,124),(7,36),(8,112),(9,34),(10,110),(11,32),(12,114),(13,127),(14,82),(15,131),(16,80),(17,129),(18,84),(19,39),(20,142),(21,37),(22,140),(23,41),(24,144),(25,107),(26,47),(27,105),(28,45),(29,103),(30,43),(31,101),(33,99),(35,97),(38,62),(40,66),(42,64),(44,73),(46,77),(48,75),(49,125),(50,86),(51,123),(52,90),(53,121),(54,88),(55,79),(56,128),(57,83),(58,132),(59,81),(60,130),(61,143),(63,141),(65,139),(67,137),(68,94),(69,135),(70,92),(71,133),(72,96),(74,108),(76,106),(78,104),(91,119),(93,117),(95,115),(98,111),(100,109),(102,113),(116,136),(118,134),(120,138)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6AB | 12A | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | C4○D12 |
| kernel | C62⋊19D4 | C12⋊Dic3 | C6.11D12 | C2×C12⋊S3 | C2×C32⋊7D4 | C2×C6×C12 | C22×C12 | C3×C12 | C62 | C2×C12 | C22×C6 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 8 | 4 | 2 | 16 | 16 | 16 |
Matrix representation of C62⋊19D4 ►in GL4(𝔽13) generated by
| 0 | 12 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 9 | 11 |
| 12 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 12 | 0 |
| 3 | 6 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 10 | 7 | 0 | 0 |
| 10 | 3 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(13))| [0,1,0,0,12,1,0,0,0,0,2,9,0,0,4,11],[12,1,0,0,12,0,0,0,0,0,1,12,0,0,1,0],[3,7,0,0,6,10,0,0,0,0,12,0,0,0,0,12],[10,10,0,0,7,3,0,0,0,0,12,1,0,0,0,1] >;
C62⋊19D4 in GAP, Magma, Sage, TeX
C_6^2\rtimes_{19}D_4 % in TeX
G:=Group("C6^2:19D4"); // GroupNames label
G:=SmallGroup(288,787);
// by ID
G=gap.SmallGroup(288,787);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1*b^3,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations