direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8⋊D9, C8⋊9D18, C24.89D6, C72⋊11C22, C18⋊1M4(2), C36.36C23, (C2×C8)⋊6D9, (C2×C72)⋊9C2, C9⋊C8⋊10C22, (C4×D9).3C4, C4.24(C4×D9), C9⋊1(C2×M4(2)), C12.74(C4×S3), C36.29(C2×C4), (C2×C24).28S3, D18.5(C2×C4), (C2×C4).99D18, C6.7(C8⋊S3), (C2×C12).411D6, (C2×Dic9).5C4, Dic9.7(C2×C4), (C22×D9).3C4, C4.36(C22×D9), C22.14(C4×D9), C18.13(C22×C4), (C4×D9).14C22, C12.197(C22×S3), (C2×C36).109C22, (C2×C9⋊C8)⋊11C2, C3.(C2×C8⋊S3), C6.52(S3×C2×C4), C2.14(C2×C4×D9), (C2×C4×D9).10C2, (C2×C6).40(C4×S3), (C2×C18).14(C2×C4), SmallGroup(288,111)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C8⋊D9
G = < a,b,c,d | a2=b8=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 384 in 102 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, C9, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, D9, C18, C18, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×M4(2), Dic9, C36, D18, D18, C2×C18, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C9⋊C8, C72, C4×D9, C2×Dic9, C2×C36, C22×D9, C2×C8⋊S3, C8⋊D9, C2×C9⋊C8, C2×C72, C2×C4×D9, C2×C8⋊D9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, D9, C4×S3, C22×S3, C2×M4(2), D18, C8⋊S3, S3×C2×C4, C4×D9, C22×D9, C2×C8⋊S3, C8⋊D9, C2×C4×D9, C2×C8⋊D9
►On 144 points
Generators in S144
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 64 28 46 10 55 19 37)(2 65 29 47 11 56 20 38)(3 66 30 48 12 57 21 39)(4 67 31 49 13 58 22 40)(5 68 32 50 14 59 23 41)(6 69 33 51 15 60 24 42)(7 70 34 52 16 61 25 43)(8 71 35 53 17 62 26 44)(9 72 36 54 18 63 27 45)(73 136 100 118 82 127 91 109)(74 137 101 119 83 128 92 110)(75 138 102 120 84 129 93 111)(76 139 103 121 85 130 94 112)(77 140 104 122 86 131 95 113)(78 141 105 123 87 132 96 114)(79 142 106 124 88 133 97 115)(80 143 107 125 89 134 98 116)(81 144 108 126 90 135 99 117)
(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 9)(2 8)(3 7)(4 6)(10 18)(11 17)(12 16)(13 15)(19 27)(20 26)(21 25)(22 24)(28 36)(29 35)(30 34)(31 33)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(73 81)(74 80)(75 79)(76 78)(82 90)(83 89)(84 88)(85 87)(91 99)(92 98)(93 97)(94 96)(100 108)(101 107)(102 106)(103 105)(109 126)(110 125)(111 124)(112 123)(113 122)(114 121)(115 120)(116 119)(117 118)(127 144)(128 143)(129 142)(130 141)(131 140)(132 139)(133 138)(134 137)(135 136)
G:=sub<Sym(144)| (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (1,64,28,46,10,55,19,37)(2,65,29,47,11,56,20,38)(3,66,30,48,12,57,21,39)(4,67,31,49,13,58,22,40)(5,68,32,50,14,59,23,41)(6,69,33,51,15,60,24,42)(7,70,34,52,16,61,25,43)(8,71,35,53,17,62,26,44)(9,72,36,54,18,63,27,45)(73,136,100,118,82,127,91,109)(74,137,101,119,83,128,92,110)(75,138,102,120,84,129,93,111)(76,139,103,121,85,130,94,112)(77,140,104,122,86,131,95,113)(78,141,105,123,87,132,96,114)(79,142,106,124,88,133,97,115)(80,143,107,125,89,134,98,116)(81,144,108,126,90,135,99,117), (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), (1,9)(2,8)(3,7)(4,6)(10,18)(11,17)(12,16)(13,15)(19,27)(20,26)(21,25)(22,24)(28,36)(29,35)(30,34)(31,33)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(73,81)(74,80)(75,79)(76,78)(82,90)(83,89)(84,88)(85,87)(91,99)(92,98)(93,97)(94,96)(100,108)(101,107)(102,106)(103,105)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(127,144)(128,143)(129,142)(130,141)(131,140)(132,139)(133,138)(134,137)(135,136)>;
G:=Group( (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (1,64,28,46,10,55,19,37)(2,65,29,47,11,56,20,38)(3,66,30,48,12,57,21,39)(4,67,31,49,13,58,22,40)(5,68,32,50,14,59,23,41)(6,69,33,51,15,60,24,42)(7,70,34,52,16,61,25,43)(8,71,35,53,17,62,26,44)(9,72,36,54,18,63,27,45)(73,136,100,118,82,127,91,109)(74,137,101,119,83,128,92,110)(75,138,102,120,84,129,93,111)(76,139,103,121,85,130,94,112)(77,140,104,122,86,131,95,113)(78,141,105,123,87,132,96,114)(79,142,106,124,88,133,97,115)(80,143,107,125,89,134,98,116)(81,144,108,126,90,135,99,117), (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), (1,9)(2,8)(3,7)(4,6)(10,18)(11,17)(12,16)(13,15)(19,27)(20,26)(21,25)(22,24)(28,36)(29,35)(30,34)(31,33)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(73,81)(74,80)(75,79)(76,78)(82,90)(83,89)(84,88)(85,87)(91,99)(92,98)(93,97)(94,96)(100,108)(101,107)(102,106)(103,105)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(127,144)(128,143)(129,142)(130,141)(131,140)(132,139)(133,138)(134,137)(135,136) );
G=PermutationGroup([[(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(1,64,28,46,10,55,19,37),(2,65,29,47,11,56,20,38),(3,66,30,48,12,57,21,39),(4,67,31,49,13,58,22,40),(5,68,32,50,14,59,23,41),(6,69,33,51,15,60,24,42),(7,70,34,52,16,61,25,43),(8,71,35,53,17,62,26,44),(9,72,36,54,18,63,27,45),(73,136,100,118,82,127,91,109),(74,137,101,119,83,128,92,110),(75,138,102,120,84,129,93,111),(76,139,103,121,85,130,94,112),(77,140,104,122,86,131,95,113),(78,141,105,123,87,132,96,114),(79,142,106,124,88,133,97,115),(80,143,107,125,89,134,98,116),(81,144,108,126,90,135,99,117)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,18),(11,17),(12,16),(13,15),(19,27),(20,26),(21,25),(22,24),(28,36),(29,35),(30,34),(31,33),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(55,72),(56,71),(57,70),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64),(73,81),(74,80),(75,79),(76,78),(82,90),(83,89),(84,88),(85,87),(91,99),(92,98),(93,97),(94,96),(100,108),(101,107),(102,106),(103,105),(109,126),(110,125),(111,124),(112,123),(113,122),(114,121),(115,120),(116,119),(117,118),(127,144),(128,143),(129,142),(130,141),(131,140),(132,139),(133,138),(134,137),(135,136)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 1 | 1 | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | D9 | C4×S3 | C4×S3 | D18 | D18 | C8⋊S3 | C4×D9 | C4×D9 | C8⋊D9 |
| kernel | C2×C8⋊D9 | C8⋊D9 | C2×C9⋊C8 | C2×C72 | C2×C4×D9 | C4×D9 | C2×Dic9 | C22×D9 | C2×C24 | C24 | C2×C12 | C18 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 4 | 3 | 2 | 2 | 6 | 3 | 8 | 6 | 6 | 24 |
Matrix representation of C2×C8⋊D9 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 72 | 0 | 0 |
| 0 | 8 | 16 |
| 0 | 57 | 65 |
| 1 | 0 | 0 |
| 0 | 45 | 42 |
| 0 | 31 | 3 |
| 72 | 0 | 0 |
| 0 | 45 | 42 |
| 0 | 70 | 28 |
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[72,0,0,0,8,57,0,16,65],[1,0,0,0,45,31,0,42,3],[72,0,0,0,45,70,0,42,28] >;
C2×C8⋊D9 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes D_9
% in TeX
G:=Group("C2xC8:D9"); // GroupNames label
G:=SmallGroup(288,111);
// by ID
G=gap.SmallGroup(288,111);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,58,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations