non-abelian, supersoluble, monomial
Aliases: C62.29D6, He3:8(C4:C4), (C6xC12).4S3, He3:3C4:4C4, (C2xHe3).6Q8, (C3xC6).9Dic6, (C2xHe3).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), (C22xHe3).22C22, C6.30(C4xC3:S3), (C2xC4xHe3).2C2, (C3xC6).23(C4xS3), (C2xC12).3(C3:S3), C2.4(C4xHe3:C2), (C3xC6).33(C3:D4), (C2xHe3).24(C2xC4), (C2xHe3:3C4).7C2, (C2xC4).1(He3:C2), C22.4(C2xHe3:C2), (C2xC6).52(C2xC3: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, C2xC4, C2xC4, C32, Dic3, C12, C2xC6, C2xC6, C4:C4, C3xC6, C2xDic3, C2xC12, C2xC12, He3, C3xDic3, C3xC12, C62, Dic3:C4, C3xC4:C4, C2xHe3, C6xDic3, C6xC12, He3:3C4, He3:3C4, C4xHe3, C22xHe3, C3xDic3:C4, C2xHe3:3C4, C2xC4xHe3, C62.29D6
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, D6, C4:C4, C3:S3, Dic6, C4xS3, C3:D4, C2xC3:S3, Dic3:C4, He3:C2, C32:4Q8, C4xC3:S3, C32:7D4, C2xHe3:C2, C6.Dic6, He3:4Q8, C4xHe3:C2, He3:7D4, C62.29D6
(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 | C4xS3 | C3:D4 | He3:C2 | C2xHe3:C2 | C4xHe3:C2 | He3:4Q8 | He3:7D4 |
kernel | C62.29D6 | C2xHe3:3C4 | C2xC4xHe3 | He3:3C4 | C6xC12 | C2xHe3 | C2xHe3 | C62 | C3xC6 | C3xC6 | C3xC6 | C2xC4 | 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