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G = C9×M5(2)  order 288 = 25·32

Direct product of C9 and M5(2)

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

Aliases: C9×M5(2), C4.C72, C1447C2, C163C18, C48.7C6, C72.6C4, C36.4C8, C8.2C36, C22.C72, C12.5C24, C24.10C12, C72.29C22, C8.8(C2×C18), (C2×C4).5C36, (C2×C6).3C24, (C2×C18).1C8, C2.3(C2×C72), (C2×C8).8C18, C3.(C3×M5(2)), (C2×C24).29C6, C18.13(C2×C8), C4.11(C2×C36), (C2×C72).18C2, C24.41(C2×C6), (C2×C36).14C4, C6.13(C2×C24), C36.49(C2×C4), (C3×M5(2)).C3, (C2×C12).19C12, C12.60(C2×C12), SmallGroup(288,60)

Series: Derived Chief Lower central Upper central

C1C2 — C9×M5(2)
C1C2C4C12C24C72C144 — C9×M5(2)
C1C2 — C9×M5(2)
C1C72 — C9×M5(2)

Generators and relations for C9×M5(2)
 G = < a,b,c | a9=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

2C2
2C6
2C18

Smallest permutation representation of C9×M5(2)
On 144 points
Generators in S144
(1 111 83 62 76 41 26 123 142)(2 112 84 63 77 42 27 124 143)(3 97 85 64 78 43 28 125 144)(4 98 86 49 79 44 29 126 129)(5 99 87 50 80 45 30 127 130)(6 100 88 51 65 46 31 128 131)(7 101 89 52 66 47 32 113 132)(8 102 90 53 67 48 17 114 133)(9 103 91 54 68 33 18 115 134)(10 104 92 55 69 34 19 116 135)(11 105 93 56 70 35 20 117 136)(12 106 94 57 71 36 21 118 137)(13 107 95 58 72 37 22 119 138)(14 108 96 59 73 38 23 120 139)(15 109 81 60 74 39 24 121 140)(16 110 82 61 75 40 25 122 141)
(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 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)(82 90)(84 92)(86 94)(88 96)(98 106)(100 108)(102 110)(104 112)(114 122)(116 124)(118 126)(120 128)(129 137)(131 139)(133 141)(135 143)

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

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

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

180 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B6C6D8A8B8C8D8E8F9A···9F12A12B12C12D12E12F16A···16H18A···18F18G···18L24A···24H24I24J24K24L36A···36L36M···36R48A···48P72A···72X72Y···72AJ144A···144AV
order1223344466668888889···912121212121216···1618···1818···1824···242424242436···3636···3648···4872···7272···72144···144
size1121111211221111221···11111222···21···12···21···122221···12···22···21···12···22···2

180 irreducible representations

dim111111111111111111111222
type+++
imageC1C2C2C3C4C4C6C6C8C8C9C12C12C18C18C24C24C36C36C72C72M5(2)C3×M5(2)C9×M5(2)
kernelC9×M5(2)C144C2×C72C3×M5(2)C72C2×C36C48C2×C24C36C2×C18M5(2)C24C2×C12C16C2×C8C12C2×C6C8C2×C4C4C22C9C3C1
# reps121222424464412688121224244824

Matrix representation of C9×M5(2) in GL2(𝔽433) generated by

2560
0256
,
345431
33488
,
10
345432
G:=sub<GL(2,GF(433))| [256,0,0,256],[345,334,431,88],[1,345,0,432] >;

C9×M5(2) in GAP, Magma, Sage, TeX

C_9\times M_5(2)
% in TeX

G:=Group("C9xM5(2)");
// GroupNames label

G:=SmallGroup(288,60);
// by ID

G=gap.SmallGroup(288,60);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-2,84,2045,142,192,124]);
// 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^9>;
// generators/relations

Export

Subgroup lattice of C9×M5(2) in TeX

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