direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C9×Q16, C8.C18, C24.3C6, C72.3C2, C18.16D4, Q8.2C18, C36.19C22, C3.(C3×Q16), C2.5(D4×C9), (C3×Q16).C3, C4.3(C2×C18), C6.16(C3×D4), (Q8×C9).2C2, (C3×Q8).7C6, C12.19(C2×C6), SmallGroup(144,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×Q16
G = < a,b,c | a9=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C9×Q16
►Regular action 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)
(1 138 53 25 42 28 59 13)(2 139 54 26 43 29 60 14)(3 140 46 27 44 30 61 15)(4 141 47 19 45 31 62 16)(5 142 48 20 37 32 63 17)(6 143 49 21 38 33 55 18)(7 144 50 22 39 34 56 10)(8 136 51 23 40 35 57 11)(9 137 52 24 41 36 58 12)(64 112 87 100 93 123 76 129)(65 113 88 101 94 124 77 130)(66 114 89 102 95 125 78 131)(67 115 90 103 96 126 79 132)(68 116 82 104 97 118 80 133)(69 117 83 105 98 119 81 134)(70 109 84 106 99 120 73 135)(71 110 85 107 91 121 74 127)(72 111 86 108 92 122 75 128)
(1 78 42 89)(2 79 43 90)(3 80 44 82)(4 81 45 83)(5 73 37 84)(6 74 38 85)(7 75 39 86)(8 76 40 87)(9 77 41 88)(10 128 22 108)(11 129 23 100)(12 130 24 101)(13 131 25 102)(14 132 26 103)(15 133 27 104)(16 134 19 105)(17 135 20 106)(18 127 21 107)(28 114 138 125)(29 115 139 126)(30 116 140 118)(31 117 141 119)(32 109 142 120)(33 110 143 121)(34 111 144 122)(35 112 136 123)(36 113 137 124)(46 97 61 68)(47 98 62 69)(48 99 63 70)(49 91 55 71)(50 92 56 72)(51 93 57 64)(52 94 58 65)(53 95 59 66)(54 96 60 67)
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), (1,138,53,25,42,28,59,13)(2,139,54,26,43,29,60,14)(3,140,46,27,44,30,61,15)(4,141,47,19,45,31,62,16)(5,142,48,20,37,32,63,17)(6,143,49,21,38,33,55,18)(7,144,50,22,39,34,56,10)(8,136,51,23,40,35,57,11)(9,137,52,24,41,36,58,12)(64,112,87,100,93,123,76,129)(65,113,88,101,94,124,77,130)(66,114,89,102,95,125,78,131)(67,115,90,103,96,126,79,132)(68,116,82,104,97,118,80,133)(69,117,83,105,98,119,81,134)(70,109,84,106,99,120,73,135)(71,110,85,107,91,121,74,127)(72,111,86,108,92,122,75,128), (1,78,42,89)(2,79,43,90)(3,80,44,82)(4,81,45,83)(5,73,37,84)(6,74,38,85)(7,75,39,86)(8,76,40,87)(9,77,41,88)(10,128,22,108)(11,129,23,100)(12,130,24,101)(13,131,25,102)(14,132,26,103)(15,133,27,104)(16,134,19,105)(17,135,20,106)(18,127,21,107)(28,114,138,125)(29,115,139,126)(30,116,140,118)(31,117,141,119)(32,109,142,120)(33,110,143,121)(34,111,144,122)(35,112,136,123)(36,113,137,124)(46,97,61,68)(47,98,62,69)(48,99,63,70)(49,91,55,71)(50,92,56,72)(51,93,57,64)(52,94,58,65)(53,95,59,66)(54,96,60,67)>;
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), (1,138,53,25,42,28,59,13)(2,139,54,26,43,29,60,14)(3,140,46,27,44,30,61,15)(4,141,47,19,45,31,62,16)(5,142,48,20,37,32,63,17)(6,143,49,21,38,33,55,18)(7,144,50,22,39,34,56,10)(8,136,51,23,40,35,57,11)(9,137,52,24,41,36,58,12)(64,112,87,100,93,123,76,129)(65,113,88,101,94,124,77,130)(66,114,89,102,95,125,78,131)(67,115,90,103,96,126,79,132)(68,116,82,104,97,118,80,133)(69,117,83,105,98,119,81,134)(70,109,84,106,99,120,73,135)(71,110,85,107,91,121,74,127)(72,111,86,108,92,122,75,128), (1,78,42,89)(2,79,43,90)(3,80,44,82)(4,81,45,83)(5,73,37,84)(6,74,38,85)(7,75,39,86)(8,76,40,87)(9,77,41,88)(10,128,22,108)(11,129,23,100)(12,130,24,101)(13,131,25,102)(14,132,26,103)(15,133,27,104)(16,134,19,105)(17,135,20,106)(18,127,21,107)(28,114,138,125)(29,115,139,126)(30,116,140,118)(31,117,141,119)(32,109,142,120)(33,110,143,121)(34,111,144,122)(35,112,136,123)(36,113,137,124)(46,97,61,68)(47,98,62,69)(48,99,63,70)(49,91,55,71)(50,92,56,72)(51,93,57,64)(52,94,58,65)(53,95,59,66)(54,96,60,67) );
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)], [(1,138,53,25,42,28,59,13),(2,139,54,26,43,29,60,14),(3,140,46,27,44,30,61,15),(4,141,47,19,45,31,62,16),(5,142,48,20,37,32,63,17),(6,143,49,21,38,33,55,18),(7,144,50,22,39,34,56,10),(8,136,51,23,40,35,57,11),(9,137,52,24,41,36,58,12),(64,112,87,100,93,123,76,129),(65,113,88,101,94,124,77,130),(66,114,89,102,95,125,78,131),(67,115,90,103,96,126,79,132),(68,116,82,104,97,118,80,133),(69,117,83,105,98,119,81,134),(70,109,84,106,99,120,73,135),(71,110,85,107,91,121,74,127),(72,111,86,108,92,122,75,128)], [(1,78,42,89),(2,79,43,90),(3,80,44,82),(4,81,45,83),(5,73,37,84),(6,74,38,85),(7,75,39,86),(8,76,40,87),(9,77,41,88),(10,128,22,108),(11,129,23,100),(12,130,24,101),(13,131,25,102),(14,132,26,103),(15,133,27,104),(16,134,19,105),(17,135,20,106),(18,127,21,107),(28,114,138,125),(29,115,139,126),(30,116,140,118),(31,117,141,119),(32,109,142,120),(33,110,143,121),(34,111,144,122),(35,112,136,123),(36,113,137,124),(46,97,61,68),(47,98,62,69),(48,99,63,70),(49,91,55,71),(50,92,56,72),(51,93,57,64),(52,94,58,65),(53,95,59,66),(54,96,60,67)]])
C9×Q16 is a maximal subgroup of
C9⋊SD32 C9⋊Q32 Q16⋊D9 D72⋊5C2
63 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 18A | ··· | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36R | 72A | ··· | 72L |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | D4 | Q16 | C3×D4 | C3×Q16 | D4×C9 | C9×Q16 |
| kernel | C9×Q16 | C72 | Q8×C9 | C3×Q16 | C24 | C3×Q8 | Q16 | C8 | Q8 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 1 | 2 | 2 | 4 | 6 | 12 |
Matrix representation of C9×Q16 ►in GL2(𝔽73) generated by
| 55 | 0 |
| 0 | 55 |
| 0 | 41 |
| 16 | 41 |
| 5 | 17 |
| 50 | 68 |
G:=sub<GL(2,GF(73))| [55,0,0,55],[0,16,41,41],[5,50,17,68] >;
C9×Q16 in GAP, Magma, Sage, TeX
C_9\times Q_{16} % in TeX
G:=Group("C9xQ16"); // GroupNames label
G:=SmallGroup(144,27);
// by ID
G=gap.SmallGroup(144,27);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-2,432,169,439,122,2019,1017,165]);
// Polycyclic
G:=Group<a,b,c|a^9=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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