direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C8○D4, D4.C36, Q8.2C36, M4(2)⋊5C18, C72.30C22, C36.55C23, (C2×C8)⋊7C18, (C2×C72)⋊15C2, C8.7(C2×C18), C4.5(C2×C36), (D4×C9).2C4, (Q8×C9).2C4, C36.34(C2×C4), C24.39(C2×C6), (C2×C24).31C6, C4○D4.5C18, (C3×D4).5C12, (C3×Q8).9C12, C12.35(C2×C12), C2.7(C22×C36), C22.1(C2×C36), (C9×M4(2))⋊11C2, C4.12(C22×C18), C12.66(C22×C6), C18.35(C22×C4), C6.35(C22×C12), (C2×C36).126C22, (C3×M4(2)).12C6, C3.(C3×C8○D4), (C2×C18).8(C2×C4), (C9×C4○D4).6C2, (C3×C8○D4).2C3, (C2×C4).25(C2×C18), (C2×C6).10(C2×C12), (C3×C4○D4).20C6, (C2×C12).143(C2×C6), SmallGroup(288,181)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C8○D4
G = < a,b,c,d | a9=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 102 in 93 conjugacy classes, 84 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, D4, Q8, C9, C12, C12, C2×C6, C2×C8, M4(2), C4○D4, C18, C18, C24, C24, C2×C12, C3×D4, C3×Q8, C8○D4, C36, C36, C2×C18, C2×C24, C3×M4(2), C3×C4○D4, C72, C72, C2×C36, D4×C9, Q8×C9, C3×C8○D4, C2×C72, C9×M4(2), C9×C4○D4, C9×C8○D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C9, C12, C2×C6, C22×C4, C18, C2×C12, C22×C6, C8○D4, C36, C2×C18, C22×C12, C2×C36, C22×C18, C3×C8○D4, C22×C36, C9×C8○D4
►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)
(1 131 59 114 51 102 37 123)(2 132 60 115 52 103 38 124)(3 133 61 116 53 104 39 125)(4 134 62 117 54 105 40 126)(5 135 63 109 46 106 41 118)(6 127 55 110 47 107 42 119)(7 128 56 111 48 108 43 120)(8 129 57 112 49 100 44 121)(9 130 58 113 50 101 45 122)(10 86 36 94 22 77 137 65)(11 87 28 95 23 78 138 66)(12 88 29 96 24 79 139 67)(13 89 30 97 25 80 140 68)(14 90 31 98 26 81 141 69)(15 82 32 99 27 73 142 70)(16 83 33 91 19 74 143 71)(17 84 34 92 20 75 144 72)(18 85 35 93 21 76 136 64)
(1 78 51 87)(2 79 52 88)(3 80 53 89)(4 81 54 90)(5 73 46 82)(6 74 47 83)(7 75 48 84)(8 76 49 85)(9 77 50 86)(10 122 22 113)(11 123 23 114)(12 124 24 115)(13 125 25 116)(14 126 26 117)(15 118 27 109)(16 119 19 110)(17 120 20 111)(18 121 21 112)(28 131 138 102)(29 132 139 103)(30 133 140 104)(31 134 141 105)(32 135 142 106)(33 127 143 107)(34 128 144 108)(35 129 136 100)(36 130 137 101)(37 95 59 66)(38 96 60 67)(39 97 61 68)(40 98 62 69)(41 99 63 70)(42 91 55 71)(43 92 56 72)(44 93 57 64)(45 94 58 65)
(1 87)(2 88)(3 89)(4 90)(5 82)(6 83)(7 84)(8 85)(9 86)(10 122)(11 123)(12 124)(13 125)(14 126)(15 118)(16 119)(17 120)(18 121)(19 110)(20 111)(21 112)(22 113)(23 114)(24 115)(25 116)(26 117)(27 109)(28 131)(29 132)(30 133)(31 134)(32 135)(33 127)(34 128)(35 129)(36 130)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 64)(45 65)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 99)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)
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), (1,131,59,114,51,102,37,123)(2,132,60,115,52,103,38,124)(3,133,61,116,53,104,39,125)(4,134,62,117,54,105,40,126)(5,135,63,109,46,106,41,118)(6,127,55,110,47,107,42,119)(7,128,56,111,48,108,43,120)(8,129,57,112,49,100,44,121)(9,130,58,113,50,101,45,122)(10,86,36,94,22,77,137,65)(11,87,28,95,23,78,138,66)(12,88,29,96,24,79,139,67)(13,89,30,97,25,80,140,68)(14,90,31,98,26,81,141,69)(15,82,32,99,27,73,142,70)(16,83,33,91,19,74,143,71)(17,84,34,92,20,75,144,72)(18,85,35,93,21,76,136,64), (1,78,51,87)(2,79,52,88)(3,80,53,89)(4,81,54,90)(5,73,46,82)(6,74,47,83)(7,75,48,84)(8,76,49,85)(9,77,50,86)(10,122,22,113)(11,123,23,114)(12,124,24,115)(13,125,25,116)(14,126,26,117)(15,118,27,109)(16,119,19,110)(17,120,20,111)(18,121,21,112)(28,131,138,102)(29,132,139,103)(30,133,140,104)(31,134,141,105)(32,135,142,106)(33,127,143,107)(34,128,144,108)(35,129,136,100)(36,130,137,101)(37,95,59,66)(38,96,60,67)(39,97,61,68)(40,98,62,69)(41,99,63,70)(42,91,55,71)(43,92,56,72)(44,93,57,64)(45,94,58,65), (1,87)(2,88)(3,89)(4,90)(5,82)(6,83)(7,84)(8,85)(9,86)(10,122)(11,123)(12,124)(13,125)(14,126)(15,118)(16,119)(17,120)(18,121)(19,110)(20,111)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,109)(28,131)(29,132)(30,133)(31,134)(32,135)(33,127)(34,128)(35,129)(36,130)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,64)(45,65)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144)>;
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), (1,131,59,114,51,102,37,123)(2,132,60,115,52,103,38,124)(3,133,61,116,53,104,39,125)(4,134,62,117,54,105,40,126)(5,135,63,109,46,106,41,118)(6,127,55,110,47,107,42,119)(7,128,56,111,48,108,43,120)(8,129,57,112,49,100,44,121)(9,130,58,113,50,101,45,122)(10,86,36,94,22,77,137,65)(11,87,28,95,23,78,138,66)(12,88,29,96,24,79,139,67)(13,89,30,97,25,80,140,68)(14,90,31,98,26,81,141,69)(15,82,32,99,27,73,142,70)(16,83,33,91,19,74,143,71)(17,84,34,92,20,75,144,72)(18,85,35,93,21,76,136,64), (1,78,51,87)(2,79,52,88)(3,80,53,89)(4,81,54,90)(5,73,46,82)(6,74,47,83)(7,75,48,84)(8,76,49,85)(9,77,50,86)(10,122,22,113)(11,123,23,114)(12,124,24,115)(13,125,25,116)(14,126,26,117)(15,118,27,109)(16,119,19,110)(17,120,20,111)(18,121,21,112)(28,131,138,102)(29,132,139,103)(30,133,140,104)(31,134,141,105)(32,135,142,106)(33,127,143,107)(34,128,144,108)(35,129,136,100)(36,130,137,101)(37,95,59,66)(38,96,60,67)(39,97,61,68)(40,98,62,69)(41,99,63,70)(42,91,55,71)(43,92,56,72)(44,93,57,64)(45,94,58,65), (1,87)(2,88)(3,89)(4,90)(5,82)(6,83)(7,84)(8,85)(9,86)(10,122)(11,123)(12,124)(13,125)(14,126)(15,118)(16,119)(17,120)(18,121)(19,110)(20,111)(21,112)(22,113)(23,114)(24,115)(25,116)(26,117)(27,109)(28,131)(29,132)(30,133)(31,134)(32,135)(33,127)(34,128)(35,129)(36,130)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,64)(45,65)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,99)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144) );
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)], [(1,131,59,114,51,102,37,123),(2,132,60,115,52,103,38,124),(3,133,61,116,53,104,39,125),(4,134,62,117,54,105,40,126),(5,135,63,109,46,106,41,118),(6,127,55,110,47,107,42,119),(7,128,56,111,48,108,43,120),(8,129,57,112,49,100,44,121),(9,130,58,113,50,101,45,122),(10,86,36,94,22,77,137,65),(11,87,28,95,23,78,138,66),(12,88,29,96,24,79,139,67),(13,89,30,97,25,80,140,68),(14,90,31,98,26,81,141,69),(15,82,32,99,27,73,142,70),(16,83,33,91,19,74,143,71),(17,84,34,92,20,75,144,72),(18,85,35,93,21,76,136,64)], [(1,78,51,87),(2,79,52,88),(3,80,53,89),(4,81,54,90),(5,73,46,82),(6,74,47,83),(7,75,48,84),(8,76,49,85),(9,77,50,86),(10,122,22,113),(11,123,23,114),(12,124,24,115),(13,125,25,116),(14,126,26,117),(15,118,27,109),(16,119,19,110),(17,120,20,111),(18,121,21,112),(28,131,138,102),(29,132,139,103),(30,133,140,104),(31,134,141,105),(32,135,142,106),(33,127,143,107),(34,128,144,108),(35,129,136,100),(36,130,137,101),(37,95,59,66),(38,96,60,67),(39,97,61,68),(40,98,62,69),(41,99,63,70),(42,91,55,71),(43,92,56,72),(44,93,57,64),(45,94,58,65)], [(1,87),(2,88),(3,89),(4,90),(5,82),(6,83),(7,84),(8,85),(9,86),(10,122),(11,123),(12,124),(13,125),(14,126),(15,118),(16,119),(17,120),(18,121),(19,110),(20,111),(21,112),(22,113),(23,114),(24,115),(25,116),(26,117),(27,109),(28,131),(29,132),(30,133),(31,134),(32,135),(33,127),(34,128),(35,129),(36,130),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,64),(45,65),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78),(52,79),(53,80),(54,81),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,99),(100,136),(101,137),(102,138),(103,139),(104,140),(105,141),(106,142),(107,143),(108,144)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6H | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 18A | ··· | 18F | 18G | ··· | 18X | 24A | ··· | 24H | 24I | ··· | 24T | 36A | ··· | 36L | 36M | ··· | 36AD | 72A | ··· | 72X | 72Y | ··· | 72BH |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 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 |
| type | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C18 | C36 | C36 | C8○D4 | C3×C8○D4 | C9×C8○D4 |
| kernel | C9×C8○D4 | C2×C72 | C9×M4(2) | C9×C4○D4 | C3×C8○D4 | D4×C9 | Q8×C9 | C2×C24 | C3×M4(2) | C3×C4○D4 | C8○D4 | C3×D4 | C3×Q8 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C9 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 2 | 6 | 6 | 2 | 6 | 12 | 4 | 18 | 18 | 6 | 36 | 12 | 4 | 8 | 24 |
Matrix representation of C9×C8○D4 ►in GL2(𝔽73) generated by
| 4 | 0 |
| 0 | 4 |
| 10 | 0 |
| 0 | 10 |
| 72 | 71 |
| 1 | 1 |
| 72 | 71 |
| 0 | 1 |
G:=sub<GL(2,GF(73))| [4,0,0,4],[10,0,0,10],[72,1,71,1],[72,0,71,1] >;
C9×C8○D4 in GAP, Magma, Sage, TeX
C_9\times C_8\circ D_4
% in TeX
G:=Group("C9xC8oD4"); // GroupNames label
G:=SmallGroup(288,181);
// by ID
G=gap.SmallGroup(288,181);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,168,1563,192,242]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations