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G = C9×C22⋊C8order 288 = 25·32

Direct product of C9 and C22⋊C8

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

Aliases: C9×C22⋊C8, C222C72, C36.65D4, C23.3C36, C18.8M4(2), (C2×C8)⋊1C18, (C2×C18)⋊1C8, (C2×C72)⋊3C2, (C2×C6).2C24, (C2×C4).3C36, (C2×C36).7C4, (C2×C24).1C6, C2.1(C2×C72), C4.16(D4×C9), C18.11(C2×C8), C6.11(C2×C24), (C2×C12).7C12, C12.82(C3×D4), (C22×C4).4C18, (C22×C6).9C12, C22.9(C2×C36), (C22×C18).3C4, (C22×C36).3C2, C2.2(C9×M4(2)), C6.8(C3×M4(2)), (C22×C12).10C6, C18.20(C22⋊C4), (C2×C36).133C22, C3.(C3×C22⋊C8), (C3×C22⋊C8).C3, C2.2(C9×C22⋊C4), (C2×C18).38(C2×C4), (C2×C6).47(C2×C12), (C2×C4).32(C2×C18), C6.20(C3×C22⋊C4), (C2×C12).167(C2×C6), SmallGroup(288,48)

Series: Derived Chief Lower central Upper central

C1C2 — C9×C22⋊C8
C1C2C6C12C2×C12C2×C36C2×C72 — C9×C22⋊C8
C1C2 — C9×C22⋊C8
C1C2×C36 — C9×C22⋊C8

Generators and relations for C9×C22⋊C8
 G = < a,b,c,d | a9=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 102 in 75 conjugacy classes, 48 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×2], C23, C9, C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], C22×C4, C18 [×3], C18 [×2], C24 [×2], C2×C12 [×2], C2×C12 [×2], C22×C6, C22⋊C8, C36 [×2], C36, C2×C18, C2×C18 [×2], C2×C18 [×2], C2×C24 [×2], C22×C12, C72 [×2], C2×C36 [×2], C2×C36 [×2], C22×C18, C3×C22⋊C8, C2×C72 [×2], C22×C36, C9×C22⋊C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C8 [×2], C2×C4, D4 [×2], C9, C12 [×2], C2×C6, C22⋊C4, C2×C8, M4(2), C18 [×3], C24 [×2], C2×C12, C3×D4 [×2], C22⋊C8, C36 [×2], C2×C18, C3×C22⋊C4, C2×C24, C3×M4(2), C72 [×2], C2×C36, D4×C9 [×2], C3×C22⋊C8, C9×C22⋊C4, C2×C72, C9×M4(2), C9×C22⋊C8

Smallest permutation representation of C9×C22⋊C8
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 77)(2 78)(3 79)(4 80)(5 81)(6 73)(7 74)(8 75)(9 76)(10 105)(11 106)(12 107)(13 108)(14 100)(15 101)(16 102)(17 103)(18 104)(19 36)(20 28)(21 29)(22 30)(23 31)(24 32)(25 33)(26 34)(27 35)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 55)(46 132)(47 133)(48 134)(49 135)(50 127)(51 128)(52 129)(53 130)(54 131)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 82)(91 113)(92 114)(93 115)(94 116)(95 117)(96 109)(97 110)(98 111)(99 112)(118 139)(119 140)(120 141)(121 142)(122 143)(123 144)(124 136)(125 137)(126 138)
(1 107)(2 108)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 81)(17 73)(18 74)(19 115)(20 116)(21 117)(22 109)(23 110)(24 111)(25 112)(26 113)(27 114)(28 94)(29 95)(30 96)(31 97)(32 98)(33 99)(34 91)(35 92)(36 93)(37 90)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 118)(53 119)(54 120)(55 70)(56 71)(57 72)(58 64)(59 65)(60 66)(61 67)(62 68)(63 69)(127 137)(128 138)(129 139)(130 140)(131 141)(132 142)(133 143)(134 144)(135 136)
(1 118 42 26 77 129 61 91)(2 119 43 27 78 130 62 92)(3 120 44 19 79 131 63 93)(4 121 45 20 80 132 55 94)(5 122 37 21 81 133 56 95)(6 123 38 22 73 134 57 96)(7 124 39 23 74 135 58 97)(8 125 40 24 75 127 59 98)(9 126 41 25 76 128 60 99)(10 137 65 32 105 50 84 111)(11 138 66 33 106 51 85 112)(12 139 67 34 107 52 86 113)(13 140 68 35 108 53 87 114)(14 141 69 36 100 54 88 115)(15 142 70 28 101 46 89 116)(16 143 71 29 102 47 90 117)(17 144 72 30 103 48 82 109)(18 136 64 31 104 49 83 110)

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

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

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

180 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F6A···6F6G6H6I6J8A···8H9A···9F12A···12H12I12J12K12L18A···18R18S···18AD24A···24P36A···36X36Y···36AJ72A···72AV
order122222334444446···666668···89···912···121212121218···1818···1824···2436···3636···3672···72
size111122111111221···122222···21···11···122221···12···22···21···12···22···2

180 irreducible representations

dim111111111111111111222222
type++++
imageC1C2C2C3C4C4C6C6C8C9C12C12C18C18C24C36C36C72D4M4(2)C3×D4C3×M4(2)D4×C9C9×M4(2)
kernelC9×C22⋊C8C2×C72C22×C36C3×C22⋊C8C2×C36C22×C18C2×C24C22×C12C2×C18C22⋊C8C2×C12C22×C6C2×C8C22×C4C2×C6C2×C4C23C22C36C18C12C6C4C2
# reps1212224286441261612124822441212

Matrix representation of C9×C22⋊C8 in GL3(𝔽73) generated by

100
0370
0037
,
100
014
0072
,
100
0720
0072
,
6300
07168
012
G:=sub<GL(3,GF(73))| [1,0,0,0,37,0,0,0,37],[1,0,0,0,1,0,0,4,72],[1,0,0,0,72,0,0,0,72],[63,0,0,0,71,1,0,68,2] >;

C9×C22⋊C8 in GAP, Magma, Sage, TeX

C_9\times C_2^2\rtimes C_8
% in TeX

G:=Group("C9xC2^2:C8");
// GroupNames label

G:=SmallGroup(288,48);
// by ID

G=gap.SmallGroup(288,48);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,268,242]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

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