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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

Aliases: C9×SD32, D8.C18, C162C18, C1446C2, C48.5C6, Q161C18, 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

C1C8 — C9×SD32
C1C2C4C12C24C72C9×Q16 — C9×SD32
C1C2C4C8 — C9×SD32
C1C18C36C72 — C9×SD32

Generators and relations for C9×SD32
 G = < a,b,c | a9=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

8C2
4C22
4C4
8C6
2Q8
2D4
4C12
4C2×C6
8C18
2C3×Q8
2C3×D4
4C36
4C2×C18
2Q8×C9
2D4×C9

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 2A2B3A3B4A4B6A6B6C6D8A8B9A···9F12A12B12C12D16A16B16C16D18A···18F18G···18L24A24B24C24D36A···36F36G···36L48A···48H72A···72L144A···144X
order12233446666889···9121212121616161618···1818···182424242436···3636···3648···4872···72144···144
size11811281188221···1228822221···18···822222···28···82···22···22···2

99 irreducible representations

dim111111111111222222222
type++++++
imageC1C2C2C2C3C6C6C6C9C18C18C18D4D8C3×D4SD32C3×D8D4×C9C3×SD32C9×D8C9×SD32
kernelC9×SD32C144C9×D8C9×Q16C3×SD32C48C3×D8C3×Q16SD32C16D8Q16C36C18C12C9C6C4C3C2C1
# reps11112222666612244681224

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

256000
025600
0010
0001
,
33010300
33033000
0024615
00209231
,
1000
043200
0010
00432432
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

Subgroup lattice of C9×SD32 in TeX

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