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G = C9×SD32order 288 = 25·32

Direct product of C9 and SD32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C9×SD32
 Chief series C1 — C2 — C4 — C12 — C24 — C72 — C9×Q16 — C9×SD32
 Lower central C1 — C2 — C4 — C8 — C9×SD32
 Upper central C1 — C18 — C36 — C72 — C9×SD32

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

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

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

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

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

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