metabelian, supersoluble, monomial
Aliases: C62.129D4, C62.250C23, (C22×C12)⋊9S3, (C2×C12).361D6, C62⋊5C4⋊14C2, C6.11D12⋊2C2, C6.Dic6⋊3C2, (C22×C6).159D6, C6.108(C4○D12), (C6×C12).290C22, C3⋊5(C23.28D6), (C2×C62).111C22, C22.9(C32⋊7D4), C2.18(C12.59D6), C32⋊22(C22.D4), (C2×C6×C12)⋊4C2, (C22×C4)⋊5(C3⋊S3), (C3×C6).276(C2×D4), C6.117(C2×C3⋊D4), C23.28(C2×C3⋊S3), C2.6(C2×C32⋊7D4), (C2×C6).97(C3⋊D4), (C3×C6).123(C4○D4), (C2×C6).267(C22×S3), (C2×C32⋊7D4).13C2, C22.55(C22×C3⋊S3), (C22×C3⋊S3).44C22, (C2×C3⋊Dic3).90C22, (C2×C4).68(C2×C3⋊S3), SmallGroup(288,786)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.129D4
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=dad=a-1b3, cbc-1=dbd=b-1, dcd=b3c-1 >
Subgroups: 844 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, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22.D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C62, C62, Dic3⋊C4, D6⋊C4, C6.D4, C2×C3⋊D4, C22×C12, C2×C3⋊Dic3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, C6×C12, C22×C3⋊S3, C2×C62, C23.28D6, C6.Dic6, C6.11D12, C62⋊5C4, C2×C32⋊7D4, C2×C6×C12, C62.129D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, C3⋊D4, C22×S3, C22.D4, C2×C3⋊S3, C4○D12, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, C23.28D6, C12.59D6, C2×C32⋊7D4, C62.129D4
►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 18 50 94 43 121)(2 13 51 95 44 122)(3 14 52 96 45 123)(4 15 53 91 46 124)(5 16 54 92 47 125)(6 17 49 93 48 126)(7 86 81 78 116 42)(8 87 82 73 117 37)(9 88 83 74 118 38)(10 89 84 75 119 39)(11 90 79 76 120 40)(12 85 80 77 115 41)(19 132 34 59 135 99)(20 127 35 60 136 100)(21 128 36 55 137 101)(22 129 31 56 138 102)(23 130 32 57 133 97)(24 131 33 58 134 98)(25 72 110 65 143 107)(26 67 111 66 144 108)(27 68 112 61 139 103)(28 69 113 62 140 104)(29 70 114 63 141 105)(30 71 109 64 142 106)
(1 112 32 115)(2 108 33 90)(3 110 34 119)(4 106 35 88)(5 114 36 117)(6 104 31 86)(7 17 140 56)(8 47 141 21)(9 15 142 60)(10 45 143 19)(11 13 144 58)(12 43 139 23)(14 72 59 75)(16 70 55 73)(18 68 57 77)(20 74 46 71)(22 78 48 69)(24 76 44 67)(25 135 84 52)(26 131 79 122)(27 133 80 50)(28 129 81 126)(29 137 82 54)(30 127 83 124)(37 125 63 128)(38 53 64 136)(39 123 65 132)(40 51 66 134)(41 121 61 130)(42 49 62 138)(85 94 103 97)(87 92 105 101)(89 96 107 99)(91 109 100 118)(93 113 102 116)(95 111 98 120)
(2 93)(3 5)(4 91)(6 95)(7 26)(8 65)(9 30)(10 63)(11 28)(12 61)(13 49)(14 125)(15 53)(16 123)(17 51)(18 121)(19 137)(20 127)(21 135)(22 131)(23 133)(24 129)(25 73)(27 77)(29 75)(31 98)(33 102)(34 36)(35 100)(37 143)(38 71)(39 141)(40 69)(41 139)(42 67)(43 50)(44 126)(45 54)(46 124)(47 52)(48 122)(55 132)(56 134)(57 130)(58 138)(59 128)(60 136)(62 76)(64 74)(66 78)(68 80)(70 84)(72 82)(79 140)(81 144)(83 142)(85 112)(86 108)(87 110)(88 106)(89 114)(90 104)(92 96)(99 101)(103 115)(105 119)(107 117)(109 118)(111 116)(113 120)
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), (1,18,50,94,43,121)(2,13,51,95,44,122)(3,14,52,96,45,123)(4,15,53,91,46,124)(5,16,54,92,47,125)(6,17,49,93,48,126)(7,86,81,78,116,42)(8,87,82,73,117,37)(9,88,83,74,118,38)(10,89,84,75,119,39)(11,90,79,76,120,40)(12,85,80,77,115,41)(19,132,34,59,135,99)(20,127,35,60,136,100)(21,128,36,55,137,101)(22,129,31,56,138,102)(23,130,32,57,133,97)(24,131,33,58,134,98)(25,72,110,65,143,107)(26,67,111,66,144,108)(27,68,112,61,139,103)(28,69,113,62,140,104)(29,70,114,63,141,105)(30,71,109,64,142,106), (1,112,32,115)(2,108,33,90)(3,110,34,119)(4,106,35,88)(5,114,36,117)(6,104,31,86)(7,17,140,56)(8,47,141,21)(9,15,142,60)(10,45,143,19)(11,13,144,58)(12,43,139,23)(14,72,59,75)(16,70,55,73)(18,68,57,77)(20,74,46,71)(22,78,48,69)(24,76,44,67)(25,135,84,52)(26,131,79,122)(27,133,80,50)(28,129,81,126)(29,137,82,54)(30,127,83,124)(37,125,63,128)(38,53,64,136)(39,123,65,132)(40,51,66,134)(41,121,61,130)(42,49,62,138)(85,94,103,97)(87,92,105,101)(89,96,107,99)(91,109,100,118)(93,113,102,116)(95,111,98,120), (2,93)(3,5)(4,91)(6,95)(7,26)(8,65)(9,30)(10,63)(11,28)(12,61)(13,49)(14,125)(15,53)(16,123)(17,51)(18,121)(19,137)(20,127)(21,135)(22,131)(23,133)(24,129)(25,73)(27,77)(29,75)(31,98)(33,102)(34,36)(35,100)(37,143)(38,71)(39,141)(40,69)(41,139)(42,67)(43,50)(44,126)(45,54)(46,124)(47,52)(48,122)(55,132)(56,134)(57,130)(58,138)(59,128)(60,136)(62,76)(64,74)(66,78)(68,80)(70,84)(72,82)(79,140)(81,144)(83,142)(85,112)(86,108)(87,110)(88,106)(89,114)(90,104)(92,96)(99,101)(103,115)(105,119)(107,117)(109,118)(111,116)(113,120)>;
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), (1,18,50,94,43,121)(2,13,51,95,44,122)(3,14,52,96,45,123)(4,15,53,91,46,124)(5,16,54,92,47,125)(6,17,49,93,48,126)(7,86,81,78,116,42)(8,87,82,73,117,37)(9,88,83,74,118,38)(10,89,84,75,119,39)(11,90,79,76,120,40)(12,85,80,77,115,41)(19,132,34,59,135,99)(20,127,35,60,136,100)(21,128,36,55,137,101)(22,129,31,56,138,102)(23,130,32,57,133,97)(24,131,33,58,134,98)(25,72,110,65,143,107)(26,67,111,66,144,108)(27,68,112,61,139,103)(28,69,113,62,140,104)(29,70,114,63,141,105)(30,71,109,64,142,106), (1,112,32,115)(2,108,33,90)(3,110,34,119)(4,106,35,88)(5,114,36,117)(6,104,31,86)(7,17,140,56)(8,47,141,21)(9,15,142,60)(10,45,143,19)(11,13,144,58)(12,43,139,23)(14,72,59,75)(16,70,55,73)(18,68,57,77)(20,74,46,71)(22,78,48,69)(24,76,44,67)(25,135,84,52)(26,131,79,122)(27,133,80,50)(28,129,81,126)(29,137,82,54)(30,127,83,124)(37,125,63,128)(38,53,64,136)(39,123,65,132)(40,51,66,134)(41,121,61,130)(42,49,62,138)(85,94,103,97)(87,92,105,101)(89,96,107,99)(91,109,100,118)(93,113,102,116)(95,111,98,120), (2,93)(3,5)(4,91)(6,95)(7,26)(8,65)(9,30)(10,63)(11,28)(12,61)(13,49)(14,125)(15,53)(16,123)(17,51)(18,121)(19,137)(20,127)(21,135)(22,131)(23,133)(24,129)(25,73)(27,77)(29,75)(31,98)(33,102)(34,36)(35,100)(37,143)(38,71)(39,141)(40,69)(41,139)(42,67)(43,50)(44,126)(45,54)(46,124)(47,52)(48,122)(55,132)(56,134)(57,130)(58,138)(59,128)(60,136)(62,76)(64,74)(66,78)(68,80)(70,84)(72,82)(79,140)(81,144)(83,142)(85,112)(86,108)(87,110)(88,106)(89,114)(90,104)(92,96)(99,101)(103,115)(105,119)(107,117)(109,118)(111,116)(113,120) );
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)], [(1,18,50,94,43,121),(2,13,51,95,44,122),(3,14,52,96,45,123),(4,15,53,91,46,124),(5,16,54,92,47,125),(6,17,49,93,48,126),(7,86,81,78,116,42),(8,87,82,73,117,37),(9,88,83,74,118,38),(10,89,84,75,119,39),(11,90,79,76,120,40),(12,85,80,77,115,41),(19,132,34,59,135,99),(20,127,35,60,136,100),(21,128,36,55,137,101),(22,129,31,56,138,102),(23,130,32,57,133,97),(24,131,33,58,134,98),(25,72,110,65,143,107),(26,67,111,66,144,108),(27,68,112,61,139,103),(28,69,113,62,140,104),(29,70,114,63,141,105),(30,71,109,64,142,106)], [(1,112,32,115),(2,108,33,90),(3,110,34,119),(4,106,35,88),(5,114,36,117),(6,104,31,86),(7,17,140,56),(8,47,141,21),(9,15,142,60),(10,45,143,19),(11,13,144,58),(12,43,139,23),(14,72,59,75),(16,70,55,73),(18,68,57,77),(20,74,46,71),(22,78,48,69),(24,76,44,67),(25,135,84,52),(26,131,79,122),(27,133,80,50),(28,129,81,126),(29,137,82,54),(30,127,83,124),(37,125,63,128),(38,53,64,136),(39,123,65,132),(40,51,66,134),(41,121,61,130),(42,49,62,138),(85,94,103,97),(87,92,105,101),(89,96,107,99),(91,109,100,118),(93,113,102,116),(95,111,98,120)], [(2,93),(3,5),(4,91),(6,95),(7,26),(8,65),(9,30),(10,63),(11,28),(12,61),(13,49),(14,125),(15,53),(16,123),(17,51),(18,121),(19,137),(20,127),(21,135),(22,131),(23,133),(24,129),(25,73),(27,77),(29,75),(31,98),(33,102),(34,36),(35,100),(37,143),(38,71),(39,141),(40,69),(41,139),(42,67),(43,50),(44,126),(45,54),(46,124),(47,52),(48,122),(55,132),(56,134),(57,130),(58,138),(59,128),(60,136),(62,76),(64,74),(66,78),(68,80),(70,84),(72,82),(79,140),(81,144),(83,142),(85,112),(86,108),(87,110),(88,106),(89,114),(90,104),(92,96),(99,101),(103,115),(105,119),(107,117),(109,118),(111,116),(113,120)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6AB | 12A | ··· | 12AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 36 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D12 |
| kernel | C62.129D4 | C6.Dic6 | C6.11D12 | C62⋊5C4 | C2×C32⋊7D4 | C2×C6×C12 | C22×C12 | C62 | C2×C12 | C22×C6 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 4 | 4 | 16 | 32 |
Matrix representation of C62.129D4 ►in GL6(𝔽13)
| 6 | 3 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 6 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 11 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 9 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(6,GF(13))| [6,10,0,0,0,0,3,7,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[5,6,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,11,11,0,0,0,0,9,2],[1,9,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;
C62.129D4 in GAP, Magma, Sage, TeX
C_6^2._{129}D_4 % in TeX
G:=Group("C6^2.129D4"); // GroupNames label
G:=SmallGroup(288,786);
// by ID
G=gap.SmallGroup(288,786);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,254,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^-1>;
// generators/relations