metabelian, supersoluble, monomial
Aliases: C62.77D6, D6⋊(C3⋊Dic3), (S3×C6)⋊2Dic3, (C3×C6).82D12, C3⋊3(D6⋊Dic3), (S3×C62).2C2, C6.30(S3×Dic3), (C32×C6).41D4, C3⋊1(C62⋊5C4), C32⋊15(D6⋊C4), C33⋊10(C22⋊C4), C2.1(C33⋊6D4), C6.5(C32⋊7D4), C2.1(C33⋊7D4), (C3×C62).7C22, C6.24(D6⋊S3), C6.26(C3⋊D12), C32⋊6(C6.D4), (S3×C3×C6)⋊4C4, (C2×C6).31S32, (S3×C2×C6).8S3, (C3×C6).91(C4×S3), (C6×C3⋊Dic3)⋊2C2, (C2×C3⋊Dic3)⋊7S3, C22.5(S3×C3⋊S3), C2.4(S3×C3⋊Dic3), C6.4(C2×C3⋊Dic3), (C2×C33⋊5C4)⋊1C2, (C22×S3).(C3⋊S3), (C3×C6).60(C3⋊D4), (C32×C6).38(C2×C4), (C3×C6).38(C2×Dic3), (C2×C6).13(C2×C3⋊S3), SmallGroup(432,449)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.77D6
G = < a,b,c,d | a6=b6=c6=1, d2=a3b3, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 1176 in 268 conjugacy classes, 82 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, C22 [×4], S3 [×2], C6 [×3], C6 [×12], C6 [×20], C2×C4 [×2], C23, C32, C32 [×4], C32 [×4], Dic3 [×17], C12, D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C2×C6 [×20], C22⋊C4, C3×S3 [×8], C3×C6 [×3], C3×C6 [×12], C3×C6 [×14], C2×Dic3 [×13], C2×C12, C22×S3, C22×C6 [×4], C33, C3×Dic3 [×4], C3⋊Dic3 [×14], S3×C6 [×8], S3×C6 [×8], C62, C62 [×4], C62 [×8], D6⋊C4, C6.D4 [×4], S3×C32 [×2], C32×C6 [×3], C6×Dic3 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×9], S3×C2×C6 [×4], C2×C62, C3×C3⋊Dic3, C33⋊5C4, S3×C3×C6 [×2], S3×C3×C6 [×2], C3×C62, D6⋊Dic3 [×4], C62⋊5C4, C6×C3⋊Dic3, C2×C33⋊5C4, S3×C62, C62.77D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×5], C2×C4, D4 [×2], Dic3 [×8], D6 [×5], C22⋊C4, C3⋊S3, C4×S3, D12, C2×Dic3 [×4], C3⋊D4 [×9], C3⋊Dic3 [×2], S32 [×4], C2×C3⋊S3, D6⋊C4, C6.D4 [×4], S3×Dic3 [×4], D6⋊S3 [×4], C3⋊D12 [×4], C2×C3⋊Dic3, C32⋊7D4 [×2], S3×C3⋊S3, D6⋊Dic3 [×4], C62⋊5C4, S3×C3⋊Dic3, C33⋊6D4, C33⋊7D4, C62.77D6
►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 48 40 33 17 57)(2 43 41 34 18 58)(3 44 42 35 13 59)(4 45 37 36 14 60)(5 46 38 31 15 55)(6 47 39 32 16 56)(7 124 143 134 19 29)(8 125 144 135 20 30)(9 126 139 136 21 25)(10 121 140 137 22 26)(11 122 141 138 23 27)(12 123 142 133 24 28)(49 92 78 99 65 68)(50 93 73 100 66 69)(51 94 74 101 61 70)(52 95 75 102 62 71)(53 96 76 97 63 72)(54 91 77 98 64 67)(79 108 86 117 111 132)(80 103 87 118 112 127)(81 104 88 119 113 128)(82 105 89 120 114 129)(83 106 90 115 109 130)(84 107 85 116 110 131)
(1 80 42 114 15 85)(2 81 37 109 16 86)(3 82 38 110 17 87)(4 83 39 111 18 88)(5 84 40 112 13 89)(6 79 41 113 14 90)(7 100 139 95 23 67)(8 101 140 96 24 68)(9 102 141 91 19 69)(10 97 142 92 20 70)(11 98 143 93 21 71)(12 99 144 94 22 72)(25 62 122 77 134 50)(26 63 123 78 135 51)(27 64 124 73 136 52)(28 65 125 74 137 53)(29 66 126 75 138 54)(30 61 121 76 133 49)(31 116 57 103 44 129)(32 117 58 104 45 130)(33 118 59 105 46 131)(34 119 60 106 47 132)(35 120 55 107 48 127)(36 115 56 108 43 128)
(1 121 36 19)(2 126 31 24)(3 125 32 23)(4 124 33 22)(5 123 34 21)(6 122 35 20)(7 17 137 45)(8 16 138 44)(9 15 133 43)(10 14 134 48)(11 13 135 47)(12 18 136 46)(25 55 142 41)(26 60 143 40)(27 59 144 39)(28 58 139 38)(29 57 140 37)(30 56 141 42)(49 111 102 105)(50 110 97 104)(51 109 98 103)(52 114 99 108)(53 113 100 107)(54 112 101 106)(61 83 91 118)(62 82 92 117)(63 81 93 116)(64 80 94 115)(65 79 95 120)(66 84 96 119)(67 127 74 90)(68 132 75 89)(69 131 76 88)(70 130 77 87)(71 129 78 86)(72 128 73 85)
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), (1,48,40,33,17,57)(2,43,41,34,18,58)(3,44,42,35,13,59)(4,45,37,36,14,60)(5,46,38,31,15,55)(6,47,39,32,16,56)(7,124,143,134,19,29)(8,125,144,135,20,30)(9,126,139,136,21,25)(10,121,140,137,22,26)(11,122,141,138,23,27)(12,123,142,133,24,28)(49,92,78,99,65,68)(50,93,73,100,66,69)(51,94,74,101,61,70)(52,95,75,102,62,71)(53,96,76,97,63,72)(54,91,77,98,64,67)(79,108,86,117,111,132)(80,103,87,118,112,127)(81,104,88,119,113,128)(82,105,89,120,114,129)(83,106,90,115,109,130)(84,107,85,116,110,131), (1,80,42,114,15,85)(2,81,37,109,16,86)(3,82,38,110,17,87)(4,83,39,111,18,88)(5,84,40,112,13,89)(6,79,41,113,14,90)(7,100,139,95,23,67)(8,101,140,96,24,68)(9,102,141,91,19,69)(10,97,142,92,20,70)(11,98,143,93,21,71)(12,99,144,94,22,72)(25,62,122,77,134,50)(26,63,123,78,135,51)(27,64,124,73,136,52)(28,65,125,74,137,53)(29,66,126,75,138,54)(30,61,121,76,133,49)(31,116,57,103,44,129)(32,117,58,104,45,130)(33,118,59,105,46,131)(34,119,60,106,47,132)(35,120,55,107,48,127)(36,115,56,108,43,128), (1,121,36,19)(2,126,31,24)(3,125,32,23)(4,124,33,22)(5,123,34,21)(6,122,35,20)(7,17,137,45)(8,16,138,44)(9,15,133,43)(10,14,134,48)(11,13,135,47)(12,18,136,46)(25,55,142,41)(26,60,143,40)(27,59,144,39)(28,58,139,38)(29,57,140,37)(30,56,141,42)(49,111,102,105)(50,110,97,104)(51,109,98,103)(52,114,99,108)(53,113,100,107)(54,112,101,106)(61,83,91,118)(62,82,92,117)(63,81,93,116)(64,80,94,115)(65,79,95,120)(66,84,96,119)(67,127,74,90)(68,132,75,89)(69,131,76,88)(70,130,77,87)(71,129,78,86)(72,128,73,85)>;
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), (1,48,40,33,17,57)(2,43,41,34,18,58)(3,44,42,35,13,59)(4,45,37,36,14,60)(5,46,38,31,15,55)(6,47,39,32,16,56)(7,124,143,134,19,29)(8,125,144,135,20,30)(9,126,139,136,21,25)(10,121,140,137,22,26)(11,122,141,138,23,27)(12,123,142,133,24,28)(49,92,78,99,65,68)(50,93,73,100,66,69)(51,94,74,101,61,70)(52,95,75,102,62,71)(53,96,76,97,63,72)(54,91,77,98,64,67)(79,108,86,117,111,132)(80,103,87,118,112,127)(81,104,88,119,113,128)(82,105,89,120,114,129)(83,106,90,115,109,130)(84,107,85,116,110,131), (1,80,42,114,15,85)(2,81,37,109,16,86)(3,82,38,110,17,87)(4,83,39,111,18,88)(5,84,40,112,13,89)(6,79,41,113,14,90)(7,100,139,95,23,67)(8,101,140,96,24,68)(9,102,141,91,19,69)(10,97,142,92,20,70)(11,98,143,93,21,71)(12,99,144,94,22,72)(25,62,122,77,134,50)(26,63,123,78,135,51)(27,64,124,73,136,52)(28,65,125,74,137,53)(29,66,126,75,138,54)(30,61,121,76,133,49)(31,116,57,103,44,129)(32,117,58,104,45,130)(33,118,59,105,46,131)(34,119,60,106,47,132)(35,120,55,107,48,127)(36,115,56,108,43,128), (1,121,36,19)(2,126,31,24)(3,125,32,23)(4,124,33,22)(5,123,34,21)(6,122,35,20)(7,17,137,45)(8,16,138,44)(9,15,133,43)(10,14,134,48)(11,13,135,47)(12,18,136,46)(25,55,142,41)(26,60,143,40)(27,59,144,39)(28,58,139,38)(29,57,140,37)(30,56,141,42)(49,111,102,105)(50,110,97,104)(51,109,98,103)(52,114,99,108)(53,113,100,107)(54,112,101,106)(61,83,91,118)(62,82,92,117)(63,81,93,116)(64,80,94,115)(65,79,95,120)(66,84,96,119)(67,127,74,90)(68,132,75,89)(69,131,76,88)(70,130,77,87)(71,129,78,86)(72,128,73,85) );
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)], [(1,48,40,33,17,57),(2,43,41,34,18,58),(3,44,42,35,13,59),(4,45,37,36,14,60),(5,46,38,31,15,55),(6,47,39,32,16,56),(7,124,143,134,19,29),(8,125,144,135,20,30),(9,126,139,136,21,25),(10,121,140,137,22,26),(11,122,141,138,23,27),(12,123,142,133,24,28),(49,92,78,99,65,68),(50,93,73,100,66,69),(51,94,74,101,61,70),(52,95,75,102,62,71),(53,96,76,97,63,72),(54,91,77,98,64,67),(79,108,86,117,111,132),(80,103,87,118,112,127),(81,104,88,119,113,128),(82,105,89,120,114,129),(83,106,90,115,109,130),(84,107,85,116,110,131)], [(1,80,42,114,15,85),(2,81,37,109,16,86),(3,82,38,110,17,87),(4,83,39,111,18,88),(5,84,40,112,13,89),(6,79,41,113,14,90),(7,100,139,95,23,67),(8,101,140,96,24,68),(9,102,141,91,19,69),(10,97,142,92,20,70),(11,98,143,93,21,71),(12,99,144,94,22,72),(25,62,122,77,134,50),(26,63,123,78,135,51),(27,64,124,73,136,52),(28,65,125,74,137,53),(29,66,126,75,138,54),(30,61,121,76,133,49),(31,116,57,103,44,129),(32,117,58,104,45,130),(33,118,59,105,46,131),(34,119,60,106,47,132),(35,120,55,107,48,127),(36,115,56,108,43,128)], [(1,121,36,19),(2,126,31,24),(3,125,32,23),(4,124,33,22),(5,123,34,21),(6,122,35,20),(7,17,137,45),(8,16,138,44),(9,15,133,43),(10,14,134,48),(11,13,135,47),(12,18,136,46),(25,55,142,41),(26,60,143,40),(27,59,144,39),(28,58,139,38),(29,57,140,37),(30,56,141,42),(49,111,102,105),(50,110,97,104),(51,109,98,103),(52,114,99,108),(53,113,100,107),(54,112,101,106),(61,83,91,118),(62,82,92,117),(63,81,93,116),(64,80,94,115),(65,79,95,120),(66,84,96,119),(67,127,74,90),(68,132,75,89),(69,131,76,88),(70,130,77,87),(71,129,78,86),(72,128,73,85)])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 4C | 4D | 6A | ··· | 6O | 6P | ··· | 6AA | 6AB | ··· | 6AQ | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | Dic3 | D6 | C4×S3 | D12 | C3⋊D4 | S32 | S3×Dic3 | D6⋊S3 | C3⋊D12 |
| kernel | C62.77D6 | C6×C3⋊Dic3 | C2×C33⋊5C4 | S3×C62 | S3×C3×C6 | C2×C3⋊Dic3 | S3×C2×C6 | C32×C6 | S3×C6 | C62 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 5 | 2 | 2 | 18 | 4 | 4 | 4 | 4 |
Matrix representation of C62.77D6 ►in GL8(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(13))| [0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,6,2,0,0,0,0,0,0,2,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[10,10,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,11,6,0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C62.77D6 in GAP, Magma, Sage, TeX
C_6^2._{77}D_6 % in TeX
G:=Group("C6^2.77D6"); // GroupNames label
G:=SmallGroup(432,449);
// by ID
G=gap.SmallGroup(432,449);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^6=1,d^2=a^3*b^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^-1>;
// generators/relations