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## G = C9⋊SD32order 288 = 25·32

### The semidirect product of C9 and SD32 acting via SD32/Q16=C2

Aliases: C93SD32, Q161D9, C36.5D4, C24.8D6, C8.6D18, D72.2C2, C18.10D8, C72.4C22, C9⋊C163C2, (C9×Q16)⋊1C2, C2.6(D4⋊D9), C4.3(C9⋊D4), C3.(C8.6D6), (C3×Q16).1S3, C6.17(D4⋊S3), C12.3(C3⋊D4), SmallGroup(288,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C9⋊SD32
 Chief series C1 — C3 — C9 — C18 — C36 — C72 — D72 — C9⋊SD32
 Lower central C9 — C18 — C36 — C72 — C9⋊SD32
 Upper central C1 — C2 — C4 — C8 — Q16

Generators and relations for C9⋊SD32
G = < a,b,c | a9=b16=c2=1, bab-1=cac=a-1, cbc=b7 >

Smallest permutation representation of C9⋊SD32
On 144 points
Generators in S144
(1 121 51 129 85 32 79 103 34)(2 35 104 80 17 86 130 52 122)(3 123 53 131 87 18 65 105 36)(4 37 106 66 19 88 132 54 124)(5 125 55 133 89 20 67 107 38)(6 39 108 68 21 90 134 56 126)(7 127 57 135 91 22 69 109 40)(8 41 110 70 23 92 136 58 128)(9 113 59 137 93 24 71 111 42)(10 43 112 72 25 94 138 60 114)(11 115 61 139 95 26 73 97 44)(12 45 98 74 27 96 140 62 116)(13 117 63 141 81 28 75 99 46)(14 47 100 76 29 82 142 64 118)(15 119 49 143 83 30 77 101 48)(16 33 102 78 31 84 144 50 120)
(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 92)(18 83)(19 90)(20 81)(21 88)(22 95)(23 86)(24 93)(25 84)(26 91)(27 82)(28 89)(29 96)(30 87)(31 94)(32 85)(33 114)(34 121)(35 128)(36 119)(37 126)(38 117)(39 124)(40 115)(41 122)(42 113)(43 120)(44 127)(45 118)(46 125)(47 116)(48 123)(49 105)(50 112)(51 103)(52 110)(53 101)(54 108)(55 99)(56 106)(57 97)(58 104)(59 111)(60 102)(61 109)(62 100)(63 107)(64 98)(65 143)(66 134)(67 141)(68 132)(69 139)(70 130)(71 137)(72 144)(73 135)(74 142)(75 133)(76 140)(77 131)(78 138)(79 129)(80 136)

G:=sub<Sym(144)| (1,121,51,129,85,32,79,103,34)(2,35,104,80,17,86,130,52,122)(3,123,53,131,87,18,65,105,36)(4,37,106,66,19,88,132,54,124)(5,125,55,133,89,20,67,107,38)(6,39,108,68,21,90,134,56,126)(7,127,57,135,91,22,69,109,40)(8,41,110,70,23,92,136,58,128)(9,113,59,137,93,24,71,111,42)(10,43,112,72,25,94,138,60,114)(11,115,61,139,95,26,73,97,44)(12,45,98,74,27,96,140,62,116)(13,117,63,141,81,28,75,99,46)(14,47,100,76,29,82,142,64,118)(15,119,49,143,83,30,77,101,48)(16,33,102,78,31,84,144,50,120), (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)(17,92)(18,83)(19,90)(20,81)(21,88)(22,95)(23,86)(24,93)(25,84)(26,91)(27,82)(28,89)(29,96)(30,87)(31,94)(32,85)(33,114)(34,121)(35,128)(36,119)(37,126)(38,117)(39,124)(40,115)(41,122)(42,113)(43,120)(44,127)(45,118)(46,125)(47,116)(48,123)(49,105)(50,112)(51,103)(52,110)(53,101)(54,108)(55,99)(56,106)(57,97)(58,104)(59,111)(60,102)(61,109)(62,100)(63,107)(64,98)(65,143)(66,134)(67,141)(68,132)(69,139)(70,130)(71,137)(72,144)(73,135)(74,142)(75,133)(76,140)(77,131)(78,138)(79,129)(80,136)>;

G:=Group( (1,121,51,129,85,32,79,103,34)(2,35,104,80,17,86,130,52,122)(3,123,53,131,87,18,65,105,36)(4,37,106,66,19,88,132,54,124)(5,125,55,133,89,20,67,107,38)(6,39,108,68,21,90,134,56,126)(7,127,57,135,91,22,69,109,40)(8,41,110,70,23,92,136,58,128)(9,113,59,137,93,24,71,111,42)(10,43,112,72,25,94,138,60,114)(11,115,61,139,95,26,73,97,44)(12,45,98,74,27,96,140,62,116)(13,117,63,141,81,28,75,99,46)(14,47,100,76,29,82,142,64,118)(15,119,49,143,83,30,77,101,48)(16,33,102,78,31,84,144,50,120), (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)(17,92)(18,83)(19,90)(20,81)(21,88)(22,95)(23,86)(24,93)(25,84)(26,91)(27,82)(28,89)(29,96)(30,87)(31,94)(32,85)(33,114)(34,121)(35,128)(36,119)(37,126)(38,117)(39,124)(40,115)(41,122)(42,113)(43,120)(44,127)(45,118)(46,125)(47,116)(48,123)(49,105)(50,112)(51,103)(52,110)(53,101)(54,108)(55,99)(56,106)(57,97)(58,104)(59,111)(60,102)(61,109)(62,100)(63,107)(64,98)(65,143)(66,134)(67,141)(68,132)(69,139)(70,130)(71,137)(72,144)(73,135)(74,142)(75,133)(76,140)(77,131)(78,138)(79,129)(80,136) );

G=PermutationGroup([[(1,121,51,129,85,32,79,103,34),(2,35,104,80,17,86,130,52,122),(3,123,53,131,87,18,65,105,36),(4,37,106,66,19,88,132,54,124),(5,125,55,133,89,20,67,107,38),(6,39,108,68,21,90,134,56,126),(7,127,57,135,91,22,69,109,40),(8,41,110,70,23,92,136,58,128),(9,113,59,137,93,24,71,111,42),(10,43,112,72,25,94,138,60,114),(11,115,61,139,95,26,73,97,44),(12,45,98,74,27,96,140,62,116),(13,117,63,141,81,28,75,99,46),(14,47,100,76,29,82,142,64,118),(15,119,49,143,83,30,77,101,48),(16,33,102,78,31,84,144,50,120)], [(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),(17,92),(18,83),(19,90),(20,81),(21,88),(22,95),(23,86),(24,93),(25,84),(26,91),(27,82),(28,89),(29,96),(30,87),(31,94),(32,85),(33,114),(34,121),(35,128),(36,119),(37,126),(38,117),(39,124),(40,115),(41,122),(42,113),(43,120),(44,127),(45,118),(46,125),(47,116),(48,123),(49,105),(50,112),(51,103),(52,110),(53,101),(54,108),(55,99),(56,106),(57,97),(58,104),(59,111),(60,102),(61,109),(62,100),(63,107),(64,98),(65,143),(66,134),(67,141),(68,132),(69,139),(70,130),(71,137),(72,144),(73,135),(74,142),(75,133),(76,140),(77,131),(78,138),(79,129),(80,136)]])

39 conjugacy classes

 class 1 2A 2B 3 4A 4B 6 8A 8B 9A 9B 9C 12A 12B 12C 16A 16B 16C 16D 18A 18B 18C 24A 24B 36A 36B 36C 36D ··· 36I 72A ··· 72F order 1 2 2 3 4 4 6 8 8 9 9 9 12 12 12 16 16 16 16 18 18 18 24 24 36 36 36 36 ··· 36 72 ··· 72 size 1 1 72 2 2 8 2 2 2 2 2 2 4 8 8 18 18 18 18 2 2 2 4 4 4 4 4 8 ··· 8 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D6 D8 D9 C3⋊D4 SD32 D18 C9⋊D4 D4⋊S3 C8.6D6 D4⋊D9 C9⋊SD32 kernel C9⋊SD32 C9⋊C16 D72 C9×Q16 C3×Q16 C36 C24 C18 Q16 C12 C9 C8 C4 C6 C3 C2 C1 # reps 1 1 1 1 1 1 1 2 3 2 4 3 6 1 2 3 6

Matrix representation of C9⋊SD32 in GL4(𝔽433) generated by

 350 36 0 0 397 386 0 0 0 0 1 0 0 0 0 1
,
 83 386 0 0 36 350 0 0 0 0 231 5 0 0 194 246
,
 350 47 0 0 397 83 0 0 0 0 1 288 0 0 0 432
G:=sub<GL(4,GF(433))| [350,397,0,0,36,386,0,0,0,0,1,0,0,0,0,1],[83,36,0,0,386,350,0,0,0,0,231,194,0,0,5,246],[350,397,0,0,47,83,0,0,0,0,1,0,0,0,288,432] >;

C9⋊SD32 in GAP, Magma, Sage, TeX

C_9\rtimes {\rm SD}_{32}
% in TeX

G:=Group("C9:SD32");
// GroupNames label

G:=SmallGroup(288,35);
// by ID

G=gap.SmallGroup(288,35);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,120,254,135,142,675,346,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^9=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^7>;
// generators/relations

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