metabelian, supersoluble, monomial
Aliases: C62.69D4, C62.230C23, (C2×C6).16D12, C6.53(C2×D12), (C2×C12).30D6, (C22×C6).92D6, C12⋊Dic3⋊8C2, C6.11D12⋊6C2, (C6×C12).15C22, C6.97(D4⋊2S3), (C2×C62).69C22, C22.4(C12⋊S3), C3⋊3(C23.21D6), C2.10(C12.D6), C32⋊18(C22.D4), (C3×C22⋊C4)⋊4S3, C22⋊C4⋊6(C3⋊S3), C2.8(C2×C12⋊S3), (C3×C6).193(C2×D4), C23.21(C2×C3⋊S3), (C32×C22⋊C4)⋊5C2, (C22×C3⋊Dic3)⋊6C2, (C3×C6).144(C4○D4), (C2×C6).247(C22×S3), (C2×C32⋊7D4).12C2, C22.45(C22×C3⋊S3), (C22×C3⋊S3).42C22, (C2×C3⋊Dic3).82C22, (C2×C4).7(C2×C3⋊S3), SmallGroup(288,743)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.69D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=ab3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 908 in 234 conjugacy classes, 77 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C22.D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C62, C62, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C2×C3⋊Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, C22×C3⋊S3, C2×C62, C23.21D6, C12⋊Dic3, C6.11D12, C32×C22⋊C4, C22×C3⋊Dic3, C2×C32⋊7D4, C62.69D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, D12, C22×S3, C22.D4, C2×C3⋊S3, C2×D12, D4⋊2S3, C12⋊S3, C22×C3⋊S3, C23.21D6, C2×C12⋊S3, C12.D6, C62.69D4
►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)
(1 119 130 110 39 133)(2 120 131 111 40 134)(3 115 132 112 41 135)(4 116 127 113 42 136)(5 117 128 114 37 137)(6 118 129 109 38 138)(7 94 77 14 101 29)(8 95 78 15 102 30)(9 96 73 16 97 25)(10 91 74 17 98 26)(11 92 75 18 99 27)(12 93 76 13 100 28)(19 33 71 82 61 121)(20 34 72 83 62 122)(21 35 67 84 63 123)(22 36 68 79 64 124)(23 31 69 80 65 125)(24 32 70 81 66 126)(43 139 107 59 53 87)(44 140 108 60 54 88)(45 141 103 55 49 89)(46 142 104 56 50 90)(47 143 105 57 51 85)(48 144 106 58 52 86)
(1 87 82 78)(2 108 83 25)(3 89 84 74)(4 104 79 27)(5 85 80 76)(6 106 81 29)(7 118 58 66)(8 39 59 33)(9 120 60 62)(10 41 55 35)(11 116 56 64)(12 37 57 31)(13 117 47 65)(14 38 48 32)(15 119 43 61)(16 40 44 34)(17 115 45 63)(18 42 46 36)(19 30 110 107)(20 73 111 88)(21 26 112 103)(22 75 113 90)(23 28 114 105)(24 77 109 86)(49 67 91 135)(50 124 92 127)(51 69 93 137)(52 126 94 129)(53 71 95 133)(54 122 96 131)(68 99 136 142)(70 101 138 144)(72 97 134 140)(98 132 141 123)(100 128 143 125)(102 130 139 121)
(1 30 110 78)(2 29 111 77)(3 28 112 76)(4 27 113 75)(5 26 114 74)(6 25 109 73)(7 131 14 134)(8 130 15 133)(9 129 16 138)(10 128 17 137)(11 127 18 136)(12 132 13 135)(19 87 82 107)(20 86 83 106)(21 85 84 105)(22 90 79 104)(23 89 80 103)(24 88 81 108)(31 49 65 141)(32 54 66 140)(33 53 61 139)(34 52 62 144)(35 51 63 143)(36 50 64 142)(37 91 117 98)(38 96 118 97)(39 95 119 102)(40 94 120 101)(41 93 115 100)(42 92 116 99)(43 71 59 121)(44 70 60 126)(45 69 55 125)(46 68 56 124)(47 67 57 123)(48 72 58 122)
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), (1,119,130,110,39,133)(2,120,131,111,40,134)(3,115,132,112,41,135)(4,116,127,113,42,136)(5,117,128,114,37,137)(6,118,129,109,38,138)(7,94,77,14,101,29)(8,95,78,15,102,30)(9,96,73,16,97,25)(10,91,74,17,98,26)(11,92,75,18,99,27)(12,93,76,13,100,28)(19,33,71,82,61,121)(20,34,72,83,62,122)(21,35,67,84,63,123)(22,36,68,79,64,124)(23,31,69,80,65,125)(24,32,70,81,66,126)(43,139,107,59,53,87)(44,140,108,60,54,88)(45,141,103,55,49,89)(46,142,104,56,50,90)(47,143,105,57,51,85)(48,144,106,58,52,86), (1,87,82,78)(2,108,83,25)(3,89,84,74)(4,104,79,27)(5,85,80,76)(6,106,81,29)(7,118,58,66)(8,39,59,33)(9,120,60,62)(10,41,55,35)(11,116,56,64)(12,37,57,31)(13,117,47,65)(14,38,48,32)(15,119,43,61)(16,40,44,34)(17,115,45,63)(18,42,46,36)(19,30,110,107)(20,73,111,88)(21,26,112,103)(22,75,113,90)(23,28,114,105)(24,77,109,86)(49,67,91,135)(50,124,92,127)(51,69,93,137)(52,126,94,129)(53,71,95,133)(54,122,96,131)(68,99,136,142)(70,101,138,144)(72,97,134,140)(98,132,141,123)(100,128,143,125)(102,130,139,121), (1,30,110,78)(2,29,111,77)(3,28,112,76)(4,27,113,75)(5,26,114,74)(6,25,109,73)(7,131,14,134)(8,130,15,133)(9,129,16,138)(10,128,17,137)(11,127,18,136)(12,132,13,135)(19,87,82,107)(20,86,83,106)(21,85,84,105)(22,90,79,104)(23,89,80,103)(24,88,81,108)(31,49,65,141)(32,54,66,140)(33,53,61,139)(34,52,62,144)(35,51,63,143)(36,50,64,142)(37,91,117,98)(38,96,118,97)(39,95,119,102)(40,94,120,101)(41,93,115,100)(42,92,116,99)(43,71,59,121)(44,70,60,126)(45,69,55,125)(46,68,56,124)(47,67,57,123)(48,72,58,122)>;
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), (1,119,130,110,39,133)(2,120,131,111,40,134)(3,115,132,112,41,135)(4,116,127,113,42,136)(5,117,128,114,37,137)(6,118,129,109,38,138)(7,94,77,14,101,29)(8,95,78,15,102,30)(9,96,73,16,97,25)(10,91,74,17,98,26)(11,92,75,18,99,27)(12,93,76,13,100,28)(19,33,71,82,61,121)(20,34,72,83,62,122)(21,35,67,84,63,123)(22,36,68,79,64,124)(23,31,69,80,65,125)(24,32,70,81,66,126)(43,139,107,59,53,87)(44,140,108,60,54,88)(45,141,103,55,49,89)(46,142,104,56,50,90)(47,143,105,57,51,85)(48,144,106,58,52,86), (1,87,82,78)(2,108,83,25)(3,89,84,74)(4,104,79,27)(5,85,80,76)(6,106,81,29)(7,118,58,66)(8,39,59,33)(9,120,60,62)(10,41,55,35)(11,116,56,64)(12,37,57,31)(13,117,47,65)(14,38,48,32)(15,119,43,61)(16,40,44,34)(17,115,45,63)(18,42,46,36)(19,30,110,107)(20,73,111,88)(21,26,112,103)(22,75,113,90)(23,28,114,105)(24,77,109,86)(49,67,91,135)(50,124,92,127)(51,69,93,137)(52,126,94,129)(53,71,95,133)(54,122,96,131)(68,99,136,142)(70,101,138,144)(72,97,134,140)(98,132,141,123)(100,128,143,125)(102,130,139,121), (1,30,110,78)(2,29,111,77)(3,28,112,76)(4,27,113,75)(5,26,114,74)(6,25,109,73)(7,131,14,134)(8,130,15,133)(9,129,16,138)(10,128,17,137)(11,127,18,136)(12,132,13,135)(19,87,82,107)(20,86,83,106)(21,85,84,105)(22,90,79,104)(23,89,80,103)(24,88,81,108)(31,49,65,141)(32,54,66,140)(33,53,61,139)(34,52,62,144)(35,51,63,143)(36,50,64,142)(37,91,117,98)(38,96,118,97)(39,95,119,102)(40,94,120,101)(41,93,115,100)(42,92,116,99)(43,71,59,121)(44,70,60,126)(45,69,55,125)(46,68,56,124)(47,67,57,123)(48,72,58,122) );
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)], [(1,119,130,110,39,133),(2,120,131,111,40,134),(3,115,132,112,41,135),(4,116,127,113,42,136),(5,117,128,114,37,137),(6,118,129,109,38,138),(7,94,77,14,101,29),(8,95,78,15,102,30),(9,96,73,16,97,25),(10,91,74,17,98,26),(11,92,75,18,99,27),(12,93,76,13,100,28),(19,33,71,82,61,121),(20,34,72,83,62,122),(21,35,67,84,63,123),(22,36,68,79,64,124),(23,31,69,80,65,125),(24,32,70,81,66,126),(43,139,107,59,53,87),(44,140,108,60,54,88),(45,141,103,55,49,89),(46,142,104,56,50,90),(47,143,105,57,51,85),(48,144,106,58,52,86)], [(1,87,82,78),(2,108,83,25),(3,89,84,74),(4,104,79,27),(5,85,80,76),(6,106,81,29),(7,118,58,66),(8,39,59,33),(9,120,60,62),(10,41,55,35),(11,116,56,64),(12,37,57,31),(13,117,47,65),(14,38,48,32),(15,119,43,61),(16,40,44,34),(17,115,45,63),(18,42,46,36),(19,30,110,107),(20,73,111,88),(21,26,112,103),(22,75,113,90),(23,28,114,105),(24,77,109,86),(49,67,91,135),(50,124,92,127),(51,69,93,137),(52,126,94,129),(53,71,95,133),(54,122,96,131),(68,99,136,142),(70,101,138,144),(72,97,134,140),(98,132,141,123),(100,128,143,125),(102,130,139,121)], [(1,30,110,78),(2,29,111,77),(3,28,112,76),(4,27,113,75),(5,26,114,74),(6,25,109,73),(7,131,14,134),(8,130,15,133),(9,129,16,138),(10,128,17,137),(11,127,18,136),(12,132,13,135),(19,87,82,107),(20,86,83,106),(21,85,84,105),(22,90,79,104),(23,89,80,103),(24,88,81,108),(31,49,65,141),(32,54,66,140),(33,53,61,139),(34,52,62,144),(35,51,63,143),(36,50,64,142),(37,91,117,98),(38,96,118,97),(39,95,119,102),(40,94,120,101),(41,93,115,100),(42,92,116,99),(43,71,59,121),(44,70,60,126),(45,69,55,125),(46,68,56,124),(47,67,57,123),(48,72,58,122)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6L | 6M | ··· | 6T | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 36 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 18 | 18 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | D12 | D4⋊2S3 |
| kernel | C62.69D4 | C12⋊Dic3 | C6.11D12 | C32×C22⋊C4 | C22×C3⋊Dic3 | C2×C32⋊7D4 | C3×C22⋊C4 | C62 | C2×C12 | C22×C6 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 4 | 4 | 16 | 8 |
Matrix representation of C62.69D4 ►in GL6(𝔽13)
| 1 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 10 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,10,0,0,0,0,7,10,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
C62.69D4 in GAP, Magma, Sage, TeX
C_6^2._{69}D_4 % in TeX
G:=Group("C6^2.69D4"); // GroupNames label
G:=SmallGroup(288,743);
// by ID
G=gap.SmallGroup(288,743);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,219,142,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=b^3,a*b=b*a,c*a*c^-1=a*b^3,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations