non-abelian, supersoluble, monomial
Aliases: C62.29D6, He3⋊8(C4⋊C4), (C6×C12).4S3, He3⋊3C4⋊4C4, (C2×He3).6Q8, (C3×C6).9Dic6, (C2×He3).32D4, C2.1(He3⋊7D4), C2.1(He3⋊4Q8), C6.9(C32⋊4Q8), C32⋊6(Dic3⋊C4), C6.29(C32⋊7D4), C3.2(C6.Dic6), (C22×He3).22C22, C6.30(C4×C3⋊S3), (C2×C4×He3).2C2, (C3×C6).23(C4×S3), (C2×C12).3(C3⋊S3), C2.4(C4×He3⋊C2), (C3×C6).33(C3⋊D4), (C2×He3).24(C2×C4), (C2×He3⋊3C4).7C2, (C2×C4).1(He3⋊C2), C22.4(C2×He3⋊C2), (C2×C6).52(C2×C3⋊S3), SmallGroup(432,187)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.29D6
G = < a,b,c,d | a6=b6=1, c6=d2=a3, ab=ba, cac-1=ab4, dad-1=a-1b2, bc=cb, bd=db, dcd-1=b3c5 >
Subgroups: 453 in 143 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C2×C12, He3, C3×Dic3, C3×C12, C62, Dic3⋊C4, C3×C4⋊C4, C2×He3, C6×Dic3, C6×C12, He3⋊3C4, He3⋊3C4, C4×He3, C22×He3, C3×Dic3⋊C4, C2×He3⋊3C4, C2×C4×He3, C62.29D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, C3⋊S3, Dic6, C4×S3, C3⋊D4, C2×C3⋊S3, Dic3⋊C4, He3⋊C2, C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, C2×He3⋊C2, C6.Dic6, He3⋊4Q8, C4×He3⋊C2, He3⋊7D4, C62.29D6
►On 144 points
Generators in S144
(1 7)(2 58 117 8 52 111)(3 112 53 9 118 59)(4 10)(5 49 120 11 55 114)(6 115 56 12 109 50)(13 71 47 19 65 41)(14 42 66 20 48 72)(15 21)(16 62 38 22 68 44)(17 45 69 23 39 63)(18 24)(25 136 90 31 142 96)(26 32)(27 86 144 33 92 138)(28 139 93 34 133 87)(29 35)(30 89 135 36 95 141)(37 43)(40 46)(51 57)(54 60)(61 67)(64 70)(73 127 98 79 121 104)(74 105 122 80 99 128)(75 81)(76 130 101 82 124 107)(77 108 125 83 102 131)(78 84)(85 91)(88 94)(97 103)(100 106)(110 116)(113 119)(123 129)(126 132)(134 140)(137 143)
(1 32 116 137 51 85)(2 33 117 138 52 86)(3 34 118 139 53 87)(4 35 119 140 54 88)(5 36 120 141 55 89)(6 25 109 142 56 90)(7 26 110 143 57 91)(8 27 111 144 58 92)(9 28 112 133 59 93)(10 29 113 134 60 94)(11 30 114 135 49 95)(12 31 115 136 50 96)(13 79 47 104 65 127)(14 80 48 105 66 128)(15 81 37 106 67 129)(16 82 38 107 68 130)(17 83 39 108 69 131)(18 84 40 97 70 132)(19 73 41 98 71 121)(20 74 42 99 72 122)(21 75 43 100 61 123)(22 76 44 101 62 124)(23 77 45 102 63 125)(24 78 46 103 64 126)
(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 14 7 20)(2 98 8 104)(3 24 9 18)(4 108 10 102)(5 22 11 16)(6 106 12 100)(13 138 19 144)(15 136 21 142)(17 134 23 140)(25 67 31 61)(26 74 32 80)(27 65 33 71)(28 84 34 78)(29 63 35 69)(30 82 36 76)(37 96 43 90)(38 120 44 114)(39 94 45 88)(40 118 46 112)(41 92 47 86)(42 116 48 110)(49 68 55 62)(50 75 56 81)(51 66 57 72)(52 73 58 79)(53 64 59 70)(54 83 60 77)(85 128 91 122)(87 126 93 132)(89 124 95 130)(97 139 103 133)(99 137 105 143)(101 135 107 141)(109 129 115 123)(111 127 117 121)(113 125 119 131)
G:=sub<Sym(144)| (1,7)(2,58,117,8,52,111)(3,112,53,9,118,59)(4,10)(5,49,120,11,55,114)(6,115,56,12,109,50)(13,71,47,19,65,41)(14,42,66,20,48,72)(15,21)(16,62,38,22,68,44)(17,45,69,23,39,63)(18,24)(25,136,90,31,142,96)(26,32)(27,86,144,33,92,138)(28,139,93,34,133,87)(29,35)(30,89,135,36,95,141)(37,43)(40,46)(51,57)(54,60)(61,67)(64,70)(73,127,98,79,121,104)(74,105,122,80,99,128)(75,81)(76,130,101,82,124,107)(77,108,125,83,102,131)(78,84)(85,91)(88,94)(97,103)(100,106)(110,116)(113,119)(123,129)(126,132)(134,140)(137,143), (1,32,116,137,51,85)(2,33,117,138,52,86)(3,34,118,139,53,87)(4,35,119,140,54,88)(5,36,120,141,55,89)(6,25,109,142,56,90)(7,26,110,143,57,91)(8,27,111,144,58,92)(9,28,112,133,59,93)(10,29,113,134,60,94)(11,30,114,135,49,95)(12,31,115,136,50,96)(13,79,47,104,65,127)(14,80,48,105,66,128)(15,81,37,106,67,129)(16,82,38,107,68,130)(17,83,39,108,69,131)(18,84,40,97,70,132)(19,73,41,98,71,121)(20,74,42,99,72,122)(21,75,43,100,61,123)(22,76,44,101,62,124)(23,77,45,102,63,125)(24,78,46,103,64,126), (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), (1,14,7,20)(2,98,8,104)(3,24,9,18)(4,108,10,102)(5,22,11,16)(6,106,12,100)(13,138,19,144)(15,136,21,142)(17,134,23,140)(25,67,31,61)(26,74,32,80)(27,65,33,71)(28,84,34,78)(29,63,35,69)(30,82,36,76)(37,96,43,90)(38,120,44,114)(39,94,45,88)(40,118,46,112)(41,92,47,86)(42,116,48,110)(49,68,55,62)(50,75,56,81)(51,66,57,72)(52,73,58,79)(53,64,59,70)(54,83,60,77)(85,128,91,122)(87,126,93,132)(89,124,95,130)(97,139,103,133)(99,137,105,143)(101,135,107,141)(109,129,115,123)(111,127,117,121)(113,125,119,131)>;
G:=Group( (1,7)(2,58,117,8,52,111)(3,112,53,9,118,59)(4,10)(5,49,120,11,55,114)(6,115,56,12,109,50)(13,71,47,19,65,41)(14,42,66,20,48,72)(15,21)(16,62,38,22,68,44)(17,45,69,23,39,63)(18,24)(25,136,90,31,142,96)(26,32)(27,86,144,33,92,138)(28,139,93,34,133,87)(29,35)(30,89,135,36,95,141)(37,43)(40,46)(51,57)(54,60)(61,67)(64,70)(73,127,98,79,121,104)(74,105,122,80,99,128)(75,81)(76,130,101,82,124,107)(77,108,125,83,102,131)(78,84)(85,91)(88,94)(97,103)(100,106)(110,116)(113,119)(123,129)(126,132)(134,140)(137,143), (1,32,116,137,51,85)(2,33,117,138,52,86)(3,34,118,139,53,87)(4,35,119,140,54,88)(5,36,120,141,55,89)(6,25,109,142,56,90)(7,26,110,143,57,91)(8,27,111,144,58,92)(9,28,112,133,59,93)(10,29,113,134,60,94)(11,30,114,135,49,95)(12,31,115,136,50,96)(13,79,47,104,65,127)(14,80,48,105,66,128)(15,81,37,106,67,129)(16,82,38,107,68,130)(17,83,39,108,69,131)(18,84,40,97,70,132)(19,73,41,98,71,121)(20,74,42,99,72,122)(21,75,43,100,61,123)(22,76,44,101,62,124)(23,77,45,102,63,125)(24,78,46,103,64,126), (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), (1,14,7,20)(2,98,8,104)(3,24,9,18)(4,108,10,102)(5,22,11,16)(6,106,12,100)(13,138,19,144)(15,136,21,142)(17,134,23,140)(25,67,31,61)(26,74,32,80)(27,65,33,71)(28,84,34,78)(29,63,35,69)(30,82,36,76)(37,96,43,90)(38,120,44,114)(39,94,45,88)(40,118,46,112)(41,92,47,86)(42,116,48,110)(49,68,55,62)(50,75,56,81)(51,66,57,72)(52,73,58,79)(53,64,59,70)(54,83,60,77)(85,128,91,122)(87,126,93,132)(89,124,95,130)(97,139,103,133)(99,137,105,143)(101,135,107,141)(109,129,115,123)(111,127,117,121)(113,125,119,131) );
G=PermutationGroup([[(1,7),(2,58,117,8,52,111),(3,112,53,9,118,59),(4,10),(5,49,120,11,55,114),(6,115,56,12,109,50),(13,71,47,19,65,41),(14,42,66,20,48,72),(15,21),(16,62,38,22,68,44),(17,45,69,23,39,63),(18,24),(25,136,90,31,142,96),(26,32),(27,86,144,33,92,138),(28,139,93,34,133,87),(29,35),(30,89,135,36,95,141),(37,43),(40,46),(51,57),(54,60),(61,67),(64,70),(73,127,98,79,121,104),(74,105,122,80,99,128),(75,81),(76,130,101,82,124,107),(77,108,125,83,102,131),(78,84),(85,91),(88,94),(97,103),(100,106),(110,116),(113,119),(123,129),(126,132),(134,140),(137,143)], [(1,32,116,137,51,85),(2,33,117,138,52,86),(3,34,118,139,53,87),(4,35,119,140,54,88),(5,36,120,141,55,89),(6,25,109,142,56,90),(7,26,110,143,57,91),(8,27,111,144,58,92),(9,28,112,133,59,93),(10,29,113,134,60,94),(11,30,114,135,49,95),(12,31,115,136,50,96),(13,79,47,104,65,127),(14,80,48,105,66,128),(15,81,37,106,67,129),(16,82,38,107,68,130),(17,83,39,108,69,131),(18,84,40,97,70,132),(19,73,41,98,71,121),(20,74,42,99,72,122),(21,75,43,100,61,123),(22,76,44,101,62,124),(23,77,45,102,63,125),(24,78,46,103,64,126)], [(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)], [(1,14,7,20),(2,98,8,104),(3,24,9,18),(4,108,10,102),(5,22,11,16),(6,106,12,100),(13,138,19,144),(15,136,21,142),(17,134,23,140),(25,67,31,61),(26,74,32,80),(27,65,33,71),(28,84,34,78),(29,63,35,69),(30,82,36,76),(37,96,43,90),(38,120,44,114),(39,94,45,88),(40,118,46,112),(41,92,47,86),(42,116,48,110),(49,68,55,62),(50,75,56,81),(51,66,57,72),(52,73,58,79),(53,64,59,70),(54,83,60,77),(85,128,91,122),(87,126,93,132),(89,124,95,130),(97,139,103,133),(99,137,105,143),(101,135,107,141),(109,129,115,123),(111,127,117,121),(113,125,119,131)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6R | 12A | 12B | 12C | 12D | 12E | ··· | 12T | 12U | ··· | 12AB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 18 | 18 | 18 | 18 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 18 | ··· | 18 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | Dic6 | C4×S3 | C3⋊D4 | He3⋊C2 | C2×He3⋊C2 | C4×He3⋊C2 | He3⋊4Q8 | He3⋊7D4 |
| kernel | C62.29D6 | C2×He3⋊3C4 | C2×C4×He3 | He3⋊3C4 | C6×C12 | C2×He3 | C2×He3 | C62 | C3×C6 | C3×C6 | C3×C6 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 1 | 1 | 4 | 8 | 8 | 8 | 4 | 4 | 8 | 2 | 2 |
Matrix representation of C62.29D6 ►in GL5(𝔽13)
| 0 | 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 3 | 12 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 4 | 2 | 0 | 0 | 0 |
| 11 | 2 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 12 | 5 | 5 |
| 1 | 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 5 | 5 |
| 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(5,GF(13))| [0,12,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,3,10,0,0,0,12,0,4],[12,0,0,0,0,0,12,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[4,11,0,0,0,2,2,0,0,0,0,0,8,0,12,0,0,0,0,5,0,0,0,8,5],[1,0,0,0,0,1,12,0,0,0,0,0,8,12,0,0,0,0,5,0,0,0,0,5,8] >;
C62.29D6 in GAP, Magma, Sage, TeX
C_6^2._{29}D_6 % in TeX
G:=Group("C6^2.29D6"); // GroupNames label
G:=SmallGroup(432,187);
// by ID
G=gap.SmallGroup(432,187);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=a^3,a*b=b*a,c*a*c^-1=a*b^4,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=b^3*c^5>;
// generators/relations