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, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C9, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C22×C4, C18, C18, C24, C2×C12, C2×C12, C22×C6, C22⋊C8, C36, C36, C2×C18, C2×C18, C2×C18, C2×C24, C22×C12, C72, C2×C36, C2×C36, C22×C18, C3×C22⋊C8, C2×C72, C22×C36, C9×C22⋊C8
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, C9, C12, C2×C6, C22⋊C4, C2×C8, M4(2), C18, C24, C2×C12, C3×D4, C22⋊C8, C36, C2×C18, C3×C22⋊C4, C2×C24, C3×M4(2), C72, C2×C36, D4×C9, 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 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 37)(17 38)(18 39)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 28)(46 132)(47 133)(48 134)(49 135)(50 127)(51 128)(52 129)(53 130)(54 131)(55 107)(56 108)(57 100)(58 101)(59 102)(60 103)(61 104)(62 105)(63 106)(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 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 64)(19 117)(20 109)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(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 77)(56 78)(57 79)(58 80)(59 81)(60 73)(61 74)(62 75)(63 76)(127 137)(128 138)(129 139)(130 140)(131 141)(132 142)(133 143)(134 144)(135 136)
(1 118 42 24 77 129 12 91)(2 119 43 25 78 130 13 92)(3 120 44 26 79 131 14 93)(4 121 45 27 80 132 15 94)(5 122 37 19 81 133 16 95)(6 123 38 20 73 134 17 96)(7 124 39 21 74 135 18 97)(8 125 40 22 75 127 10 98)(9 126 41 23 76 128 11 99)(28 101 46 89 116 58 142 70)(29 102 47 90 117 59 143 71)(30 103 48 82 109 60 144 72)(31 104 49 83 110 61 136 64)(32 105 50 84 111 62 137 65)(33 106 51 85 112 63 138 66)(34 107 52 86 113 55 139 67)(35 108 53 87 114 56 140 68)(36 100 54 88 115 57 141 69)
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)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,37)(17,38)(18,39)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,28)(46,132)(47,133)(48,134)(49,135)(50,127)(51,128)(52,129)(53,130)(54,131)(55,107)(56,108)(57,100)(58,101)(59,102)(60,103)(61,104)(62,105)(63,106)(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)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,64)(19,117)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(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)(55,77)(56,78)(57,79)(58,80)(59,81)(60,73)(61,74)(62,75)(63,76)(127,137)(128,138)(129,139)(130,140)(131,141)(132,142)(133,143)(134,144)(135,136), (1,118,42,24,77,129,12,91)(2,119,43,25,78,130,13,92)(3,120,44,26,79,131,14,93)(4,121,45,27,80,132,15,94)(5,122,37,19,81,133,16,95)(6,123,38,20,73,134,17,96)(7,124,39,21,74,135,18,97)(8,125,40,22,75,127,10,98)(9,126,41,23,76,128,11,99)(28,101,46,89,116,58,142,70)(29,102,47,90,117,59,143,71)(30,103,48,82,109,60,144,72)(31,104,49,83,110,61,136,64)(32,105,50,84,111,62,137,65)(33,106,51,85,112,63,138,66)(34,107,52,86,113,55,139,67)(35,108,53,87,114,56,140,68)(36,100,54,88,115,57,141,69)>;
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)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,37)(17,38)(18,39)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,28)(46,132)(47,133)(48,134)(49,135)(50,127)(51,128)(52,129)(53,130)(54,131)(55,107)(56,108)(57,100)(58,101)(59,102)(60,103)(61,104)(62,105)(63,106)(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)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,64)(19,117)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(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)(55,77)(56,78)(57,79)(58,80)(59,81)(60,73)(61,74)(62,75)(63,76)(127,137)(128,138)(129,139)(130,140)(131,141)(132,142)(133,143)(134,144)(135,136), (1,118,42,24,77,129,12,91)(2,119,43,25,78,130,13,92)(3,120,44,26,79,131,14,93)(4,121,45,27,80,132,15,94)(5,122,37,19,81,133,16,95)(6,123,38,20,73,134,17,96)(7,124,39,21,74,135,18,97)(8,125,40,22,75,127,10,98)(9,126,41,23,76,128,11,99)(28,101,46,89,116,58,142,70)(29,102,47,90,117,59,143,71)(30,103,48,82,109,60,144,72)(31,104,49,83,110,61,136,64)(32,105,50,84,111,62,137,65)(33,106,51,85,112,63,138,66)(34,107,52,86,113,55,139,67)(35,108,53,87,114,56,140,68)(36,100,54,88,115,57,141,69) );
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),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,37),(17,38),(18,39),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,28),(46,132),(47,133),(48,134),(49,135),(50,127),(51,128),(52,129),(53,130),(54,131),(55,107),(56,108),(57,100),(58,101),(59,102),(60,103),(61,104),(62,105),(63,106),(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),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,64),(19,117),(20,109),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(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),(55,77),(56,78),(57,79),(58,80),(59,81),(60,73),(61,74),(62,75),(63,76),(127,137),(128,138),(129,139),(130,140),(131,141),(132,142),(133,143),(134,144),(135,136)], [(1,118,42,24,77,129,12,91),(2,119,43,25,78,130,13,92),(3,120,44,26,79,131,14,93),(4,121,45,27,80,132,15,94),(5,122,37,19,81,133,16,95),(6,123,38,20,73,134,17,96),(7,124,39,21,74,135,18,97),(8,125,40,22,75,127,10,98),(9,126,41,23,76,128,11,99),(28,101,46,89,116,58,142,70),(29,102,47,90,117,59,143,71),(30,103,48,82,109,60,144,72),(31,104,49,83,110,61,136,64),(32,105,50,84,111,62,137,65),(33,106,51,85,112,63,138,66),(34,107,52,86,113,55,139,67),(35,108,53,87,114,56,140,68),(36,100,54,88,115,57,141,69)]])
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