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G = C9×Q16order 144 = 24·32

Direct product of C9 and Q16

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

 Derived series C1 — C4 — C9×Q16
 Chief series C1 — C2 — C6 — C12 — C36 — Q8×C9 — C9×Q16
 Lower central C1 — C2 — C4 — C9×Q16
 Upper central C1 — C18 — C36 — C9×Q16

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  D725C2

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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