direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C9⋊C8, C18⋊C8, C36.3C4, C4.14D18, C12.52D6, C4.3Dic9, C12.5Dic3, C36.14C22, C22.2Dic9, C9⋊2(C2×C8), C6.2(C3⋊C8), (C2×C4).5D9, (C2×C18).2C4, (C2×C36).5C2, C18.6(C2×C4), (C2×C12).13S3, C2.1(C2×Dic9), C6.6(C2×Dic3), (C2×C6).3Dic3, C3.(C2×C3⋊C8), SmallGroup(144,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C2×C9⋊C8 |
Generators and relations for C2×C9⋊C8
G = < a,b,c | a2=b9=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C9⋊C8
►Regular action on 144 points
Generators in S144
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 37)(8 38)(9 39)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)
(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)
(1 104 31 86 13 95 22 77)(2 103 32 85 14 94 23 76)(3 102 33 84 15 93 24 75)(4 101 34 83 16 92 25 74)(5 100 35 82 17 91 26 73)(6 108 36 90 18 99 27 81)(7 107 28 89 10 98 19 80)(8 106 29 88 11 97 20 79)(9 105 30 87 12 96 21 78)(37 143 64 125 46 134 55 116)(38 142 65 124 47 133 56 115)(39 141 66 123 48 132 57 114)(40 140 67 122 49 131 58 113)(41 139 68 121 50 130 59 112)(42 138 69 120 51 129 60 111)(43 137 70 119 52 128 61 110)(44 136 71 118 53 127 62 109)(45 144 72 126 54 135 63 117)
G:=sub<Sym(144)| (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,37)(8,38)(9,39)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144), (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), (1,104,31,86,13,95,22,77)(2,103,32,85,14,94,23,76)(3,102,33,84,15,93,24,75)(4,101,34,83,16,92,25,74)(5,100,35,82,17,91,26,73)(6,108,36,90,18,99,27,81)(7,107,28,89,10,98,19,80)(8,106,29,88,11,97,20,79)(9,105,30,87,12,96,21,78)(37,143,64,125,46,134,55,116)(38,142,65,124,47,133,56,115)(39,141,66,123,48,132,57,114)(40,140,67,122,49,131,58,113)(41,139,68,121,50,130,59,112)(42,138,69,120,51,129,60,111)(43,137,70,119,52,128,61,110)(44,136,71,118,53,127,62,109)(45,144,72,126,54,135,63,117)>;
G:=Group( (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,37)(8,38)(9,39)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,121)(86,122)(87,123)(88,124)(89,125)(90,126)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144), (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), (1,104,31,86,13,95,22,77)(2,103,32,85,14,94,23,76)(3,102,33,84,15,93,24,75)(4,101,34,83,16,92,25,74)(5,100,35,82,17,91,26,73)(6,108,36,90,18,99,27,81)(7,107,28,89,10,98,19,80)(8,106,29,88,11,97,20,79)(9,105,30,87,12,96,21,78)(37,143,64,125,46,134,55,116)(38,142,65,124,47,133,56,115)(39,141,66,123,48,132,57,114)(40,140,67,122,49,131,58,113)(41,139,68,121,50,130,59,112)(42,138,69,120,51,129,60,111)(43,137,70,119,52,128,61,110)(44,136,71,118,53,127,62,109)(45,144,72,126,54,135,63,117) );
G=PermutationGroup([[(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,37),(8,38),(9,39),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(73,109),(74,110),(75,111),(76,112),(77,113),(78,114),(79,115),(80,116),(81,117),(82,118),(83,119),(84,120),(85,121),(86,122),(87,123),(88,124),(89,125),(90,126),(91,127),(92,128),(93,129),(94,130),(95,131),(96,132),(97,133),(98,134),(99,135),(100,136),(101,137),(102,138),(103,139),(104,140),(105,141),(106,142),(107,143),(108,144)], [(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)], [(1,104,31,86,13,95,22,77),(2,103,32,85,14,94,23,76),(3,102,33,84,15,93,24,75),(4,101,34,83,16,92,25,74),(5,100,35,82,17,91,26,73),(6,108,36,90,18,99,27,81),(7,107,28,89,10,98,19,80),(8,106,29,88,11,97,20,79),(9,105,30,87,12,96,21,78),(37,143,64,125,46,134,55,116),(38,142,65,124,47,133,56,115),(39,141,66,123,48,132,57,114),(40,140,67,122,49,131,58,113),(41,139,68,121,50,130,59,112),(42,138,69,120,51,129,60,111),(43,137,70,119,52,128,61,110),(44,136,71,118,53,127,62,109),(45,144,72,126,54,135,63,117)]])
C2×C9⋊C8 is a maximal subgroup of
C42.D9 C36⋊C8 C36.Q8 C4.Dic18 C18.Q16 C18.D8 C8×Dic9 Dic9⋊C8 C72⋊C4 D18⋊C8 C36.53D4 C36.55D4 D4⋊Dic9 Q8⋊2Dic9 C2×C8×D9 D36.C4 D4.Dic9 D4.9D18
C2×C9⋊C8 is a maximal quotient of
C36⋊C8 C36.C8 C36.55D4
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | ··· | 8H | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | Dic3 | D6 | Dic3 | D9 | C3⋊C8 | Dic9 | D18 | Dic9 | C9⋊C8 |
| kernel | C2×C9⋊C8 | C9⋊C8 | C2×C36 | C36 | C2×C18 | C18 | C2×C12 | C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 3 | 3 | 12 |
Matrix representation of C2×C9⋊C8 ►in GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 1 | 0 | 0 |
| 0 | 45 | 42 |
| 0 | 31 | 3 |
| 72 | 0 | 0 |
| 0 | 2 | 31 |
| 0 | 29 | 71 |
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[1,0,0,0,45,31,0,42,3],[72,0,0,0,2,29,0,31,71] >;
C2×C9⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_9\rtimes C_8
% in TeX
G:=Group("C2xC9:C8"); // GroupNames label
G:=SmallGroup(144,9);
// by ID
G=gap.SmallGroup(144,9);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,50,2404,208,3461]);
// Polycyclic
G:=Group<a,b,c|a^2=b^9=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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