direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C9×SD32, D8.C18, C16⋊2C18, C144⋊6C2, C48.5C6, Q16⋊1C18, C36.38D4, C18.16D8, C72.25C22, C4.2(D4×C9), C2.4(C9×D8), C3.(C3×SD32), C8.3(C2×C18), (C9×Q16)⋊5C2, (C3×D8).3C6, (C9×D8).2C2, C6.16(C3×D8), (C3×SD32).C3, C24.22(C2×C6), C12.37(C3×D4), (C3×Q16).2C6, SmallGroup(288,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×SD32
G = < a,b,c | a9=b16=c2=1, ab=ba, ac=ca, cbc=b7 >
Smallest permutation representation of C9×SD32
►On 144 points
Generators in S144
(1 110 92 137 60 31 114 46 66)(2 111 93 138 61 32 115 47 67)(3 112 94 139 62 17 116 48 68)(4 97 95 140 63 18 117 33 69)(5 98 96 141 64 19 118 34 70)(6 99 81 142 49 20 119 35 71)(7 100 82 143 50 21 120 36 72)(8 101 83 144 51 22 121 37 73)(9 102 84 129 52 23 122 38 74)(10 103 85 130 53 24 123 39 75)(11 104 86 131 54 25 124 40 76)(12 105 87 132 55 26 125 41 77)(13 106 88 133 56 27 126 42 78)(14 107 89 134 57 28 127 43 79)(15 108 90 135 58 29 128 44 80)(16 109 91 136 59 30 113 45 65)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 29)(18 20)(19 27)(21 25)(22 32)(24 30)(26 28)(33 35)(34 42)(36 40)(37 47)(39 45)(41 43)(44 48)(49 63)(50 54)(51 61)(53 59)(55 57)(56 64)(58 62)(65 75)(67 73)(68 80)(69 71)(70 78)(72 76)(77 79)(81 95)(82 86)(83 93)(85 91)(87 89)(88 96)(90 94)(97 99)(98 106)(100 104)(101 111)(103 109)(105 107)(108 112)(113 123)(115 121)(116 128)(117 119)(118 126)(120 124)(125 127)(130 136)(131 143)(132 134)(133 141)(135 139)(138 144)(140 142)
G:=sub<Sym(144)| (1,110,92,137,60,31,114,46,66)(2,111,93,138,61,32,115,47,67)(3,112,94,139,62,17,116,48,68)(4,97,95,140,63,18,117,33,69)(5,98,96,141,64,19,118,34,70)(6,99,81,142,49,20,119,35,71)(7,100,82,143,50,21,120,36,72)(8,101,83,144,51,22,121,37,73)(9,102,84,129,52,23,122,38,74)(10,103,85,130,53,24,123,39,75)(11,104,86,131,54,25,124,40,76)(12,105,87,132,55,26,125,41,77)(13,106,88,133,56,27,126,42,78)(14,107,89,134,57,28,127,43,79)(15,108,90,135,58,29,128,44,80)(16,109,91,136,59,30,113,45,65), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,29)(18,20)(19,27)(21,25)(22,32)(24,30)(26,28)(33,35)(34,42)(36,40)(37,47)(39,45)(41,43)(44,48)(49,63)(50,54)(51,61)(53,59)(55,57)(56,64)(58,62)(65,75)(67,73)(68,80)(69,71)(70,78)(72,76)(77,79)(81,95)(82,86)(83,93)(85,91)(87,89)(88,96)(90,94)(97,99)(98,106)(100,104)(101,111)(103,109)(105,107)(108,112)(113,123)(115,121)(116,128)(117,119)(118,126)(120,124)(125,127)(130,136)(131,143)(132,134)(133,141)(135,139)(138,144)(140,142)>;
G:=Group( (1,110,92,137,60,31,114,46,66)(2,111,93,138,61,32,115,47,67)(3,112,94,139,62,17,116,48,68)(4,97,95,140,63,18,117,33,69)(5,98,96,141,64,19,118,34,70)(6,99,81,142,49,20,119,35,71)(7,100,82,143,50,21,120,36,72)(8,101,83,144,51,22,121,37,73)(9,102,84,129,52,23,122,38,74)(10,103,85,130,53,24,123,39,75)(11,104,86,131,54,25,124,40,76)(12,105,87,132,55,26,125,41,77)(13,106,88,133,56,27,126,42,78)(14,107,89,134,57,28,127,43,79)(15,108,90,135,58,29,128,44,80)(16,109,91,136,59,30,113,45,65), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,29)(18,20)(19,27)(21,25)(22,32)(24,30)(26,28)(33,35)(34,42)(36,40)(37,47)(39,45)(41,43)(44,48)(49,63)(50,54)(51,61)(53,59)(55,57)(56,64)(58,62)(65,75)(67,73)(68,80)(69,71)(70,78)(72,76)(77,79)(81,95)(82,86)(83,93)(85,91)(87,89)(88,96)(90,94)(97,99)(98,106)(100,104)(101,111)(103,109)(105,107)(108,112)(113,123)(115,121)(116,128)(117,119)(118,126)(120,124)(125,127)(130,136)(131,143)(132,134)(133,141)(135,139)(138,144)(140,142) );
G=PermutationGroup([[(1,110,92,137,60,31,114,46,66),(2,111,93,138,61,32,115,47,67),(3,112,94,139,62,17,116,48,68),(4,97,95,140,63,18,117,33,69),(5,98,96,141,64,19,118,34,70),(6,99,81,142,49,20,119,35,71),(7,100,82,143,50,21,120,36,72),(8,101,83,144,51,22,121,37,73),(9,102,84,129,52,23,122,38,74),(10,103,85,130,53,24,123,39,75),(11,104,86,131,54,25,124,40,76),(12,105,87,132,55,26,125,41,77),(13,106,88,133,56,27,126,42,78),(14,107,89,134,57,28,127,43,79),(15,108,90,135,58,29,128,44,80),(16,109,91,136,59,30,113,45,65)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,29),(18,20),(19,27),(21,25),(22,32),(24,30),(26,28),(33,35),(34,42),(36,40),(37,47),(39,45),(41,43),(44,48),(49,63),(50,54),(51,61),(53,59),(55,57),(56,64),(58,62),(65,75),(67,73),(68,80),(69,71),(70,78),(72,76),(77,79),(81,95),(82,86),(83,93),(85,91),(87,89),(88,96),(90,94),(97,99),(98,106),(100,104),(101,111),(103,109),(105,107),(108,112),(113,123),(115,121),(116,128),(117,119),(118,126),(120,124),(125,127),(130,136),(131,143),(132,134),(133,141),(135,139),(138,144),(140,142)]])
99 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 16A | 16B | 16C | 16D | 18A | ··· | 18F | 18G | ··· | 18L | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36L | 48A | ··· | 48H | 72A | ··· | 72L | 144A | ··· | 144X |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 144 | ··· | 144 |
| size | 1 | 1 | 8 | 1 | 1 | 2 | 8 | 1 | 1 | 8 | 8 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | D4 | D8 | C3×D4 | SD32 | C3×D8 | D4×C9 | C3×SD32 | C9×D8 | C9×SD32 |
| kernel | C9×SD32 | C144 | C9×D8 | C9×Q16 | C3×SD32 | C48 | C3×D8 | C3×Q16 | SD32 | C16 | D8 | Q16 | C36 | C18 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 2 | 2 | 4 | 4 | 6 | 8 | 12 | 24 |
Matrix representation of C9×SD32 ►in GL4(𝔽433) generated by
| 256 | 0 | 0 | 0 |
| 0 | 256 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 330 | 103 | 0 | 0 |
| 330 | 330 | 0 | 0 |
| 0 | 0 | 246 | 15 |
| 0 | 0 | 209 | 231 |
| 1 | 0 | 0 | 0 |
| 0 | 432 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 432 | 432 |
G:=sub<GL(4,GF(433))| [256,0,0,0,0,256,0,0,0,0,1,0,0,0,0,1],[330,330,0,0,103,330,0,0,0,0,246,209,0,0,15,231],[1,0,0,0,0,432,0,0,0,0,1,432,0,0,0,432] >;
C9×SD32 in GAP, Magma, Sage, TeX
C_9\times {\rm SD}_{32} % in TeX
G:=Group("C9xSD32"); // GroupNames label
G:=SmallGroup(288,62);
// by ID
G=gap.SmallGroup(288,62);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-2,1008,197,142,2355,1186,528,9077,4548,124]);
// Polycyclic
G:=Group<a,b,c|a^9=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations
Export