direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C22⋊C8, C22⋊2C72, 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
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
►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 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 9A | ··· | 9F | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 18A | ··· | 18R | 18S | ··· | 18AD | 24A | ··· | 24P | 36A | ··· | 36X | 36Y | ··· | 36AJ | 72A | ··· | 72AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C9 | C12 | C12 | C18 | C18 | C24 | C36 | C36 | C72 | D4 | M4(2) | C3×D4 | C3×M4(2) | D4×C9 | C9×M4(2) |
| kernel | C9×C22⋊C8 | C2×C72 | C22×C36 | C3×C22⋊C8 | C2×C36 | C22×C18 | C2×C24 | C22×C12 | C2×C18 | C22⋊C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 | C36 | C18 | C12 | C6 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 6 | 4 | 4 | 12 | 6 | 16 | 12 | 12 | 48 | 2 | 2 | 4 | 4 | 12 | 12 |
Matrix representation of C9×C22⋊C8 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 37 | 0 |
| 0 | 0 | 37 |
| 1 | 0 | 0 |
| 0 | 1 | 4 |
| 0 | 0 | 72 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 63 | 0 | 0 |
| 0 | 71 | 68 |
| 0 | 1 | 2 |
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