metabelian, supersoluble, monomial
Aliases: C62.72D4, C62.253C23, (C6×D4).29S3, (C2×C12).250D6, (C22×C6).94D6, C62⋊5C4⋊16C2, (C6×C12).266C22, C6.Dic6⋊25C2, (C2×C62).71C22, C6.105(D4⋊2S3), C3⋊5(C23.23D6), C22.4(C32⋊7D4), C2.15(C12.D6), C32⋊24(C22.D4), (D4×C3×C6).17C2, (C2×D4).5(C3⋊S3), (C3×C6).281(C2×D4), C6.122(C2×C3⋊D4), C23.12(C2×C3⋊S3), (C2×C6).25(C3⋊D4), (C22×C3⋊Dic3)⋊9C2, C2.11(C2×C32⋊7D4), (C3×C6).151(C4○D4), (C2×C6).270(C22×S3), C22.57(C22×C3⋊S3), (C2×C3⋊Dic3).165C22, (C2×C4).18(C2×C3⋊S3), SmallGroup(288,792)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C62.72D4 |
Generators and relations for C62.72D4
G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=a-1b3, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 716 in 234 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C3×D4, C22×C6, C22.D4, C3⋊Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C6.D4, C22×Dic3, C6×D4, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, D4×C32, C2×C62, C23.23D6, C6.Dic6, C62⋊5C4, C62⋊5C4, C22×C3⋊Dic3, D4×C3×C6, C62.72D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊S3, C3⋊D4, C22×S3, C22.D4, C2×C3⋊S3, D4⋊2S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, C23.23D6, C12.D6, C2×C32⋊7D4, C62.72D4
►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 38 110 23 91 97)(2 39 111 24 92 98)(3 40 112 19 93 99)(4 41 113 20 94 100)(5 42 114 21 95 101)(6 37 109 22 96 102)(7 115 105 58 65 124)(8 116 106 59 66 125)(9 117 107 60 61 126)(10 118 108 55 62 121)(11 119 103 56 63 122)(12 120 104 57 64 123)(13 67 46 33 53 27)(14 68 47 34 54 28)(15 69 48 35 49 29)(16 70 43 36 50 30)(17 71 44 31 51 25)(18 72 45 32 52 26)(73 139 137 83 88 128)(74 140 138 84 89 129)(75 141 133 79 90 130)(76 142 134 80 85 131)(77 143 135 81 86 132)(78 144 136 82 87 127)
(1 81 27 104)(2 76 28 122)(3 79 29 108)(4 74 30 126)(5 83 25 106)(6 78 26 124)(7 102 144 52)(8 114 139 71)(9 100 140 50)(10 112 141 69)(11 98 142 54)(12 110 143 67)(13 120 38 135)(14 63 39 131)(15 118 40 133)(16 61 41 129)(17 116 42 137)(18 65 37 127)(19 75 48 121)(20 84 43 107)(21 73 44 125)(22 82 45 105)(23 77 46 123)(24 80 47 103)(31 66 95 128)(32 115 96 136)(33 64 91 132)(34 119 92 134)(35 62 93 130)(36 117 94 138)(49 55 99 90)(51 59 101 88)(53 57 97 86)(56 111 85 68)(58 109 87 72)(60 113 89 70)
(1 123 23 104)(2 122 24 103)(3 121 19 108)(4 126 20 107)(5 125 21 106)(6 124 22 105)(7 109 58 102)(8 114 59 101)(9 113 60 100)(10 112 55 99)(11 111 56 98)(12 110 57 97)(13 132 33 135)(14 131 34 134)(15 130 35 133)(16 129 36 138)(17 128 31 137)(18 127 32 136)(25 73 44 83)(26 78 45 82)(27 77 46 81)(28 76 47 80)(29 75 48 79)(30 74 43 84)(37 65 96 115)(38 64 91 120)(39 63 92 119)(40 62 93 118)(41 61 94 117)(42 66 95 116)(49 141 69 90)(50 140 70 89)(51 139 71 88)(52 144 72 87)(53 143 67 86)(54 142 68 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), (1,38,110,23,91,97)(2,39,111,24,92,98)(3,40,112,19,93,99)(4,41,113,20,94,100)(5,42,114,21,95,101)(6,37,109,22,96,102)(7,115,105,58,65,124)(8,116,106,59,66,125)(9,117,107,60,61,126)(10,118,108,55,62,121)(11,119,103,56,63,122)(12,120,104,57,64,123)(13,67,46,33,53,27)(14,68,47,34,54,28)(15,69,48,35,49,29)(16,70,43,36,50,30)(17,71,44,31,51,25)(18,72,45,32,52,26)(73,139,137,83,88,128)(74,140,138,84,89,129)(75,141,133,79,90,130)(76,142,134,80,85,131)(77,143,135,81,86,132)(78,144,136,82,87,127), (1,81,27,104)(2,76,28,122)(3,79,29,108)(4,74,30,126)(5,83,25,106)(6,78,26,124)(7,102,144,52)(8,114,139,71)(9,100,140,50)(10,112,141,69)(11,98,142,54)(12,110,143,67)(13,120,38,135)(14,63,39,131)(15,118,40,133)(16,61,41,129)(17,116,42,137)(18,65,37,127)(19,75,48,121)(20,84,43,107)(21,73,44,125)(22,82,45,105)(23,77,46,123)(24,80,47,103)(31,66,95,128)(32,115,96,136)(33,64,91,132)(34,119,92,134)(35,62,93,130)(36,117,94,138)(49,55,99,90)(51,59,101,88)(53,57,97,86)(56,111,85,68)(58,109,87,72)(60,113,89,70), (1,123,23,104)(2,122,24,103)(3,121,19,108)(4,126,20,107)(5,125,21,106)(6,124,22,105)(7,109,58,102)(8,114,59,101)(9,113,60,100)(10,112,55,99)(11,111,56,98)(12,110,57,97)(13,132,33,135)(14,131,34,134)(15,130,35,133)(16,129,36,138)(17,128,31,137)(18,127,32,136)(25,73,44,83)(26,78,45,82)(27,77,46,81)(28,76,47,80)(29,75,48,79)(30,74,43,84)(37,65,96,115)(38,64,91,120)(39,63,92,119)(40,62,93,118)(41,61,94,117)(42,66,95,116)(49,141,69,90)(50,140,70,89)(51,139,71,88)(52,144,72,87)(53,143,67,86)(54,142,68,85)>;
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), (1,38,110,23,91,97)(2,39,111,24,92,98)(3,40,112,19,93,99)(4,41,113,20,94,100)(5,42,114,21,95,101)(6,37,109,22,96,102)(7,115,105,58,65,124)(8,116,106,59,66,125)(9,117,107,60,61,126)(10,118,108,55,62,121)(11,119,103,56,63,122)(12,120,104,57,64,123)(13,67,46,33,53,27)(14,68,47,34,54,28)(15,69,48,35,49,29)(16,70,43,36,50,30)(17,71,44,31,51,25)(18,72,45,32,52,26)(73,139,137,83,88,128)(74,140,138,84,89,129)(75,141,133,79,90,130)(76,142,134,80,85,131)(77,143,135,81,86,132)(78,144,136,82,87,127), (1,81,27,104)(2,76,28,122)(3,79,29,108)(4,74,30,126)(5,83,25,106)(6,78,26,124)(7,102,144,52)(8,114,139,71)(9,100,140,50)(10,112,141,69)(11,98,142,54)(12,110,143,67)(13,120,38,135)(14,63,39,131)(15,118,40,133)(16,61,41,129)(17,116,42,137)(18,65,37,127)(19,75,48,121)(20,84,43,107)(21,73,44,125)(22,82,45,105)(23,77,46,123)(24,80,47,103)(31,66,95,128)(32,115,96,136)(33,64,91,132)(34,119,92,134)(35,62,93,130)(36,117,94,138)(49,55,99,90)(51,59,101,88)(53,57,97,86)(56,111,85,68)(58,109,87,72)(60,113,89,70), (1,123,23,104)(2,122,24,103)(3,121,19,108)(4,126,20,107)(5,125,21,106)(6,124,22,105)(7,109,58,102)(8,114,59,101)(9,113,60,100)(10,112,55,99)(11,111,56,98)(12,110,57,97)(13,132,33,135)(14,131,34,134)(15,130,35,133)(16,129,36,138)(17,128,31,137)(18,127,32,136)(25,73,44,83)(26,78,45,82)(27,77,46,81)(28,76,47,80)(29,75,48,79)(30,74,43,84)(37,65,96,115)(38,64,91,120)(39,63,92,119)(40,62,93,118)(41,61,94,117)(42,66,95,116)(49,141,69,90)(50,140,70,89)(51,139,71,88)(52,144,72,87)(53,143,67,86)(54,142,68,85) );
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)], [(1,38,110,23,91,97),(2,39,111,24,92,98),(3,40,112,19,93,99),(4,41,113,20,94,100),(5,42,114,21,95,101),(6,37,109,22,96,102),(7,115,105,58,65,124),(8,116,106,59,66,125),(9,117,107,60,61,126),(10,118,108,55,62,121),(11,119,103,56,63,122),(12,120,104,57,64,123),(13,67,46,33,53,27),(14,68,47,34,54,28),(15,69,48,35,49,29),(16,70,43,36,50,30),(17,71,44,31,51,25),(18,72,45,32,52,26),(73,139,137,83,88,128),(74,140,138,84,89,129),(75,141,133,79,90,130),(76,142,134,80,85,131),(77,143,135,81,86,132),(78,144,136,82,87,127)], [(1,81,27,104),(2,76,28,122),(3,79,29,108),(4,74,30,126),(5,83,25,106),(6,78,26,124),(7,102,144,52),(8,114,139,71),(9,100,140,50),(10,112,141,69),(11,98,142,54),(12,110,143,67),(13,120,38,135),(14,63,39,131),(15,118,40,133),(16,61,41,129),(17,116,42,137),(18,65,37,127),(19,75,48,121),(20,84,43,107),(21,73,44,125),(22,82,45,105),(23,77,46,123),(24,80,47,103),(31,66,95,128),(32,115,96,136),(33,64,91,132),(34,119,92,134),(35,62,93,130),(36,117,94,138),(49,55,99,90),(51,59,101,88),(53,57,97,86),(56,111,85,68),(58,109,87,72),(60,113,89,70)], [(1,123,23,104),(2,122,24,103),(3,121,19,108),(4,126,20,107),(5,125,21,106),(6,124,22,105),(7,109,58,102),(8,114,59,101),(9,113,60,100),(10,112,55,99),(11,111,56,98),(12,110,57,97),(13,132,33,135),(14,131,34,134),(15,130,35,133),(16,129,36,138),(17,128,31,137),(18,127,32,136),(25,73,44,83),(26,78,45,82),(27,77,46,81),(28,76,47,80),(29,75,48,79),(30,74,43,84),(37,65,96,115),(38,64,91,120),(39,63,92,119),(40,62,93,118),(41,61,94,117),(42,66,95,116),(49,141,69,90),(50,140,70,89),(51,139,71,88),(52,144,72,87),(53,143,67,86),(54,142,68,85)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6L | 6M | ··· | 6AB | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 18 | 18 | 18 | 18 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | D4⋊2S3 |
| kernel | C62.72D4 | C6.Dic6 | C62⋊5C4 | C22×C3⋊Dic3 | D4×C3×C6 | C6×D4 | C62 | C2×C12 | C22×C6 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 2 | 3 | 1 | 1 | 4 | 2 | 4 | 8 | 4 | 16 | 8 |
Matrix representation of C62.72D4 ►in GL6(𝔽13)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 11 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,8,0,0,0,0,0,12,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,11,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C62.72D4 in GAP, Magma, Sage, TeX
C_6^2._{72}D_4 % in TeX
G:=Group("C6^2.72D4"); // GroupNames label
G:=SmallGroup(288,792);
// by ID
G=gap.SmallGroup(288,792);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,219,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=b^3,a*b=b*a,c*a*c^-1=a^-1*b^3,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