non-abelian, supersoluble, monomial
Aliases: C62.30D6, C4⋊(He3⋊3C4), He3⋊9(C4⋊C4), (C4×He3)⋊4C4, (C6×C12).11S3, (C3×C12)⋊2Dic3, (C3×C6).26D12, (C2×He3).7Q8, (C2×He3).25D4, (C3×C6).10Dic6, C2.1(He3⋊5D4), C32⋊5(C4⋊Dic3), C2.2(He3⋊4Q8), C6.19(C12⋊S3), C12.10(C3⋊Dic3), C6.10(C32⋊4Q8), C3.2(C12⋊Dic3), (C22×He3).23C22, (C2×C4×He3).7C2, C6.26(C2×C3⋊Dic3), C2.4(C2×He3⋊3C4), (C2×C12).21(C3⋊S3), (C2×He3).33(C2×C4), (C2×He3⋊3C4).2C2, (C3×C6).19(C2×Dic3), (C2×C4).3(He3⋊C2), C22.5(C2×He3⋊C2), (C2×C6).53(C2×C3⋊S3), SmallGroup(432,188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.30D6
G = < a,b,c,d | a6=b6=1, c6=b3, d2=a3, ab=ba, cac-1=ab2, dad-1=a-1b4, bc=cb, bd=db, dcd-1=b3c5 >
Subgroups: 453 in 143 conjugacy classes, 59 normal (19 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×C12, C2×C12, He3, C3×Dic3, C3×C12, C62, C4⋊Dic3, C3×C4⋊C4, C2×He3, C6×Dic3, C6×C12, He3⋊3C4, C4×He3, C22×He3, C3×C4⋊Dic3, C2×He3⋊3C4, C2×C4×He3, C62.30D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, C3⋊S3, Dic6, D12, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C4⋊Dic3, He3⋊C2, C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, He3⋊3C4, C2×He3⋊C2, C12⋊Dic3, He3⋊4Q8, He3⋊5D4, C2×He3⋊3C4, C62.30D6
►On 144 points
Generators in S144
(1 66 52 13 123 115)(2 63 49 14 132 112)(3 72 58 15 129 109)(4 69 55 16 126 118)(5 136 107 35 89 76)(6 133 104 36 86 73)(7 142 101 33 95 82)(8 139 98 34 92 79)(9 90 81 18 137 100)(10 87 78 19 134 97)(11 96 75 20 143 106)(12 93 84 17 140 103)(21 85 80 45 144 99)(22 94 77 46 141 108)(23 91 74 47 138 105)(24 88 83 48 135 102)(25 65 59 39 122 110)(26 62 56 40 131 119)(27 71 53 37 128 116)(28 68 50 38 125 113)(29 64 54 44 121 117)(30 61 51 41 130 114)(31 70 60 42 127 111)(32 67 57 43 124 120)
(1 29 26 3 31 28)(2 30 27 4 32 25)(5 19 45 7 17 47)(6 20 46 8 18 48)(9 24 36 11 22 34)(10 21 33 12 23 35)(13 44 40 15 42 38)(14 41 37 16 43 39)(49 51 53 55 57 59)(50 52 54 56 58 60)(61 71 69 67 65 63)(62 72 70 68 66 64)(73 75 77 79 81 83)(74 76 78 80 82 84)(85 95 93 91 89 87)(86 96 94 92 90 88)(97 99 101 103 105 107)(98 100 102 104 106 108)(109 111 113 115 117 119)(110 112 114 116 118 120)(121 131 129 127 125 123)(122 132 130 128 126 124)(133 143 141 139 137 135)(134 144 142 140 138 136)
(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 21 13 45)(2 24 14 48)(3 23 15 47)(4 22 16 46)(5 31 35 42)(6 30 36 41)(7 29 33 44)(8 32 34 43)(9 39 18 25)(10 38 19 28)(11 37 20 27)(12 40 17 26)(49 143 112 96)(50 142 113 95)(51 141 114 94)(52 140 115 93)(53 139 116 92)(54 138 117 91)(55 137 118 90)(56 136 119 89)(57 135 120 88)(58 134 109 87)(59 133 110 86)(60 144 111 85)(61 100 130 81)(62 99 131 80)(63 98 132 79)(64 97 121 78)(65 108 122 77)(66 107 123 76)(67 106 124 75)(68 105 125 74)(69 104 126 73)(70 103 127 84)(71 102 128 83)(72 101 129 82)
G:=sub<Sym(144)| (1,66,52,13,123,115)(2,63,49,14,132,112)(3,72,58,15,129,109)(4,69,55,16,126,118)(5,136,107,35,89,76)(6,133,104,36,86,73)(7,142,101,33,95,82)(8,139,98,34,92,79)(9,90,81,18,137,100)(10,87,78,19,134,97)(11,96,75,20,143,106)(12,93,84,17,140,103)(21,85,80,45,144,99)(22,94,77,46,141,108)(23,91,74,47,138,105)(24,88,83,48,135,102)(25,65,59,39,122,110)(26,62,56,40,131,119)(27,71,53,37,128,116)(28,68,50,38,125,113)(29,64,54,44,121,117)(30,61,51,41,130,114)(31,70,60,42,127,111)(32,67,57,43,124,120), (1,29,26,3,31,28)(2,30,27,4,32,25)(5,19,45,7,17,47)(6,20,46,8,18,48)(9,24,36,11,22,34)(10,21,33,12,23,35)(13,44,40,15,42,38)(14,41,37,16,43,39)(49,51,53,55,57,59)(50,52,54,56,58,60)(61,71,69,67,65,63)(62,72,70,68,66,64)(73,75,77,79,81,83)(74,76,78,80,82,84)(85,95,93,91,89,87)(86,96,94,92,90,88)(97,99,101,103,105,107)(98,100,102,104,106,108)(109,111,113,115,117,119)(110,112,114,116,118,120)(121,131,129,127,125,123)(122,132,130,128,126,124)(133,143,141,139,137,135)(134,144,142,140,138,136), (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), (1,21,13,45)(2,24,14,48)(3,23,15,47)(4,22,16,46)(5,31,35,42)(6,30,36,41)(7,29,33,44)(8,32,34,43)(9,39,18,25)(10,38,19,28)(11,37,20,27)(12,40,17,26)(49,143,112,96)(50,142,113,95)(51,141,114,94)(52,140,115,93)(53,139,116,92)(54,138,117,91)(55,137,118,90)(56,136,119,89)(57,135,120,88)(58,134,109,87)(59,133,110,86)(60,144,111,85)(61,100,130,81)(62,99,131,80)(63,98,132,79)(64,97,121,78)(65,108,122,77)(66,107,123,76)(67,106,124,75)(68,105,125,74)(69,104,126,73)(70,103,127,84)(71,102,128,83)(72,101,129,82)>;
G:=Group( (1,66,52,13,123,115)(2,63,49,14,132,112)(3,72,58,15,129,109)(4,69,55,16,126,118)(5,136,107,35,89,76)(6,133,104,36,86,73)(7,142,101,33,95,82)(8,139,98,34,92,79)(9,90,81,18,137,100)(10,87,78,19,134,97)(11,96,75,20,143,106)(12,93,84,17,140,103)(21,85,80,45,144,99)(22,94,77,46,141,108)(23,91,74,47,138,105)(24,88,83,48,135,102)(25,65,59,39,122,110)(26,62,56,40,131,119)(27,71,53,37,128,116)(28,68,50,38,125,113)(29,64,54,44,121,117)(30,61,51,41,130,114)(31,70,60,42,127,111)(32,67,57,43,124,120), (1,29,26,3,31,28)(2,30,27,4,32,25)(5,19,45,7,17,47)(6,20,46,8,18,48)(9,24,36,11,22,34)(10,21,33,12,23,35)(13,44,40,15,42,38)(14,41,37,16,43,39)(49,51,53,55,57,59)(50,52,54,56,58,60)(61,71,69,67,65,63)(62,72,70,68,66,64)(73,75,77,79,81,83)(74,76,78,80,82,84)(85,95,93,91,89,87)(86,96,94,92,90,88)(97,99,101,103,105,107)(98,100,102,104,106,108)(109,111,113,115,117,119)(110,112,114,116,118,120)(121,131,129,127,125,123)(122,132,130,128,126,124)(133,143,141,139,137,135)(134,144,142,140,138,136), (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), (1,21,13,45)(2,24,14,48)(3,23,15,47)(4,22,16,46)(5,31,35,42)(6,30,36,41)(7,29,33,44)(8,32,34,43)(9,39,18,25)(10,38,19,28)(11,37,20,27)(12,40,17,26)(49,143,112,96)(50,142,113,95)(51,141,114,94)(52,140,115,93)(53,139,116,92)(54,138,117,91)(55,137,118,90)(56,136,119,89)(57,135,120,88)(58,134,109,87)(59,133,110,86)(60,144,111,85)(61,100,130,81)(62,99,131,80)(63,98,132,79)(64,97,121,78)(65,108,122,77)(66,107,123,76)(67,106,124,75)(68,105,125,74)(69,104,126,73)(70,103,127,84)(71,102,128,83)(72,101,129,82) );
G=PermutationGroup([[(1,66,52,13,123,115),(2,63,49,14,132,112),(3,72,58,15,129,109),(4,69,55,16,126,118),(5,136,107,35,89,76),(6,133,104,36,86,73),(7,142,101,33,95,82),(8,139,98,34,92,79),(9,90,81,18,137,100),(10,87,78,19,134,97),(11,96,75,20,143,106),(12,93,84,17,140,103),(21,85,80,45,144,99),(22,94,77,46,141,108),(23,91,74,47,138,105),(24,88,83,48,135,102),(25,65,59,39,122,110),(26,62,56,40,131,119),(27,71,53,37,128,116),(28,68,50,38,125,113),(29,64,54,44,121,117),(30,61,51,41,130,114),(31,70,60,42,127,111),(32,67,57,43,124,120)], [(1,29,26,3,31,28),(2,30,27,4,32,25),(5,19,45,7,17,47),(6,20,46,8,18,48),(9,24,36,11,22,34),(10,21,33,12,23,35),(13,44,40,15,42,38),(14,41,37,16,43,39),(49,51,53,55,57,59),(50,52,54,56,58,60),(61,71,69,67,65,63),(62,72,70,68,66,64),(73,75,77,79,81,83),(74,76,78,80,82,84),(85,95,93,91,89,87),(86,96,94,92,90,88),(97,99,101,103,105,107),(98,100,102,104,106,108),(109,111,113,115,117,119),(110,112,114,116,118,120),(121,131,129,127,125,123),(122,132,130,128,126,124),(133,143,141,139,137,135),(134,144,142,140,138,136)], [(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)], [(1,21,13,45),(2,24,14,48),(3,23,15,47),(4,22,16,46),(5,31,35,42),(6,30,36,41),(7,29,33,44),(8,32,34,43),(9,39,18,25),(10,38,19,28),(11,37,20,27),(12,40,17,26),(49,143,112,96),(50,142,113,95),(51,141,114,94),(52,140,115,93),(53,139,116,92),(54,138,117,91),(55,137,118,90),(56,136,119,89),(57,135,120,88),(58,134,109,87),(59,133,110,86),(60,144,111,85),(61,100,130,81),(62,99,131,80),(63,98,132,79),(64,97,121,78),(65,108,122,77),(66,107,123,76),(67,106,124,75),(68,105,125,74),(69,104,126,73),(70,103,127,84),(71,102,128,83),(72,101,129,82)]])
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 | Dic3 | D6 | Dic6 | D12 | He3⋊C2 | He3⋊3C4 | C2×He3⋊C2 | He3⋊4Q8 | He3⋊5D4 |
| kernel | C62.30D6 | C2×He3⋊3C4 | C2×C4×He3 | C4×He3 | C6×C12 | C2×He3 | C2×He3 | C3×C12 | C62 | C3×C6 | C3×C6 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 1 | 1 | 8 | 4 | 8 | 8 | 4 | 8 | 4 | 2 | 2 |
Matrix representation of C62.30D6 ►in GL7(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 6 | 3 | 0 | 0 | 0 | 0 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 |
| 3 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 11 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 10 | 0 |
G:=sub<GL(7,GF(13))| [0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9],[6,10,0,0,0,0,0,3,3,0,0,0,0,0,0,0,3,3,0,0,0,0,0,10,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,9],[7,3,0,0,0,0,0,10,6,0,0,0,0,0,0,0,2,9,0,0,0,0,0,11,11,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,0,0,0,0,0,4,0] >;
C62.30D6 in GAP, Magma, Sage, TeX
C_6^2._{30}D_6 % in TeX
G:=Group("C6^2.30D6"); // GroupNames label
G:=SmallGroup(432,188);
// by ID
G=gap.SmallGroup(432,188);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,64,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^3,d^2=a^3,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^4,b*c=c*b,b*d=d*b,d*c*d^-1=b^3*c^5>;
// generators/relations