direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C18×C3⋊C8, C6⋊C72, C12.3C36, C36.78D6, C62.17C12, C36.11Dic3, C3⋊2(C2×C72), (C3×C18)⋊1C8, (C2×C6).5C36, (C3×C36).2C4, (C6×C18).1C4, C6.5(C2×C36), (C6×C36).1C2, (C3×C6).8C24, C4.14(S3×C18), (C2×C12).6C18, (C6×C12).39C6, (C2×C36).21S3, C4.3(C9×Dic3), C12.118(S3×C6), C12.14(C2×C18), (C3×C12).20C12, C32.3(C2×C24), C2.1(Dic3×C18), (C2×C18).9Dic3, C6.29(C6×Dic3), (C3×C36).52C22, C12.23(C3×Dic3), C18.17(C2×Dic3), C22.2(C9×Dic3), (C6×C3⋊C8).C3, (C3×C9)⋊7(C2×C8), C6.9(C3×C3⋊C8), C3.4(C6×C3⋊C8), (C3×C3⋊C8).10C6, (C2×C4).5(S3×C9), (C2×C12).48(C3×S3), (C3×C12).89(C2×C6), (C3×C18).27(C2×C4), (C3×C6).51(C2×C12), (C2×C6).26(C3×Dic3), SmallGroup(432,126)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C18×C3⋊C8 |
Generators and relations for C18×C3⋊C8
G = < a,b,c | a18=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 116 in 82 conjugacy classes, 57 normal (39 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C9, C9, C32, C12, C12, C2×C6, C2×C6, C2×C8, C18, C18, C18, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C3×C9, C36, C36, C2×C18, C2×C18, C3×C12, C62, C2×C3⋊C8, C2×C24, C3×C18, C3×C18, C72, C2×C36, C2×C36, C3×C3⋊C8, C6×C12, C3×C36, C6×C18, C2×C72, C6×C3⋊C8, C9×C3⋊C8, C6×C36, C18×C3⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C9, Dic3, C12, D6, C2×C6, C2×C8, C18, C3×S3, C3⋊C8, C24, C2×Dic3, C2×C12, C36, C2×C18, C3×Dic3, S3×C6, C2×C3⋊C8, C2×C24, S3×C9, C72, C2×C36, C3×C3⋊C8, C6×Dic3, C9×Dic3, S3×C18, C2×C72, C6×C3⋊C8, C9×C3⋊C8, Dic3×C18, C18×C3⋊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 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 67 61)(56 68 62)(57 69 63)(58 70 64)(59 71 65)(60 72 66)(73 79 85)(74 80 86)(75 81 87)(76 82 88)(77 83 89)(78 84 90)(91 103 97)(92 104 98)(93 105 99)(94 106 100)(95 107 101)(96 108 102)(109 115 121)(110 116 122)(111 117 123)(112 118 124)(113 119 125)(114 120 126)(127 139 133)(128 140 134)(129 141 135)(130 142 136)(131 143 137)(132 144 138)
(1 100 21 46 124 130 76 62)(2 101 22 47 125 131 77 63)(3 102 23 48 126 132 78 64)(4 103 24 49 109 133 79 65)(5 104 25 50 110 134 80 66)(6 105 26 51 111 135 81 67)(7 106 27 52 112 136 82 68)(8 107 28 53 113 137 83 69)(9 108 29 54 114 138 84 70)(10 91 30 37 115 139 85 71)(11 92 31 38 116 140 86 72)(12 93 32 39 117 141 87 55)(13 94 33 40 118 142 88 56)(14 95 34 41 119 143 89 57)(15 96 35 42 120 144 90 58)(16 97 36 43 121 127 73 59)(17 98 19 44 122 128 74 60)(18 99 20 45 123 129 75 61)
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), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,67,61)(56,68,62)(57,69,63)(58,70,64)(59,71,65)(60,72,66)(73,79,85)(74,80,86)(75,81,87)(76,82,88)(77,83,89)(78,84,90)(91,103,97)(92,104,98)(93,105,99)(94,106,100)(95,107,101)(96,108,102)(109,115,121)(110,116,122)(111,117,123)(112,118,124)(113,119,125)(114,120,126)(127,139,133)(128,140,134)(129,141,135)(130,142,136)(131,143,137)(132,144,138), (1,100,21,46,124,130,76,62)(2,101,22,47,125,131,77,63)(3,102,23,48,126,132,78,64)(4,103,24,49,109,133,79,65)(5,104,25,50,110,134,80,66)(6,105,26,51,111,135,81,67)(7,106,27,52,112,136,82,68)(8,107,28,53,113,137,83,69)(9,108,29,54,114,138,84,70)(10,91,30,37,115,139,85,71)(11,92,31,38,116,140,86,72)(12,93,32,39,117,141,87,55)(13,94,33,40,118,142,88,56)(14,95,34,41,119,143,89,57)(15,96,35,42,120,144,90,58)(16,97,36,43,121,127,73,59)(17,98,19,44,122,128,74,60)(18,99,20,45,123,129,75,61)>;
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), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,67,61)(56,68,62)(57,69,63)(58,70,64)(59,71,65)(60,72,66)(73,79,85)(74,80,86)(75,81,87)(76,82,88)(77,83,89)(78,84,90)(91,103,97)(92,104,98)(93,105,99)(94,106,100)(95,107,101)(96,108,102)(109,115,121)(110,116,122)(111,117,123)(112,118,124)(113,119,125)(114,120,126)(127,139,133)(128,140,134)(129,141,135)(130,142,136)(131,143,137)(132,144,138), (1,100,21,46,124,130,76,62)(2,101,22,47,125,131,77,63)(3,102,23,48,126,132,78,64)(4,103,24,49,109,133,79,65)(5,104,25,50,110,134,80,66)(6,105,26,51,111,135,81,67)(7,106,27,52,112,136,82,68)(8,107,28,53,113,137,83,69)(9,108,29,54,114,138,84,70)(10,91,30,37,115,139,85,71)(11,92,31,38,116,140,86,72)(12,93,32,39,117,141,87,55)(13,94,33,40,118,142,88,56)(14,95,34,41,119,143,89,57)(15,96,35,42,120,144,90,58)(16,97,36,43,121,127,73,59)(17,98,19,44,122,128,74,60)(18,99,20,45,123,129,75,61) );
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)], [(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36),(37,49,43),(38,50,44),(39,51,45),(40,52,46),(41,53,47),(42,54,48),(55,67,61),(56,68,62),(57,69,63),(58,70,64),(59,71,65),(60,72,66),(73,79,85),(74,80,86),(75,81,87),(76,82,88),(77,83,89),(78,84,90),(91,103,97),(92,104,98),(93,105,99),(94,106,100),(95,107,101),(96,108,102),(109,115,121),(110,116,122),(111,117,123),(112,118,124),(113,119,125),(114,120,126),(127,139,133),(128,140,134),(129,141,135),(130,142,136),(131,143,137),(132,144,138)], [(1,100,21,46,124,130,76,62),(2,101,22,47,125,131,77,63),(3,102,23,48,126,132,78,64),(4,103,24,49,109,133,79,65),(5,104,25,50,110,134,80,66),(6,105,26,51,111,135,81,67),(7,106,27,52,112,136,82,68),(8,107,28,53,113,137,83,69),(9,108,29,54,114,138,84,70),(10,91,30,37,115,139,85,71),(11,92,31,38,116,140,86,72),(12,93,32,39,117,141,87,55),(13,94,33,40,118,142,88,56),(14,95,34,41,119,143,89,57),(15,96,35,42,120,144,90,58),(16,97,36,43,121,127,73,59),(17,98,19,44,122,128,74,60),(18,99,20,45,123,129,75,61)]])
216 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 8A | ··· | 8H | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12H | 12I | ··· | 12T | 18A | ··· | 18R | 18S | ··· | 18AJ | 24A | ··· | 24P | 36A | ··· | 36X | 36Y | ··· | 36AV | 72A | ··· | 72AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
216 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 | 2 | 2 | 2 | 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 | S3 | Dic3 | D6 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | S3×C6 | C3×Dic3 | S3×C9 | C3×C3⋊C8 | C9×Dic3 | S3×C18 | C9×Dic3 | C9×C3⋊C8 |
| kernel | C18×C3⋊C8 | C9×C3⋊C8 | C6×C36 | C6×C3⋊C8 | C3×C36 | C6×C18 | C3×C3⋊C8 | C6×C12 | C3×C18 | C2×C3⋊C8 | C3×C12 | C62 | C3⋊C8 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 | C2×C36 | C36 | C36 | C2×C18 | C2×C12 | C18 | C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 6 | 4 | 4 | 12 | 6 | 16 | 12 | 12 | 48 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 6 | 8 | 6 | 6 | 6 | 24 |
Matrix representation of C18×C3⋊C8 ►in GL4(𝔽73) generated by
| 16 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 8 |
| 10 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [16,0,0,0,0,18,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,8],[10,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0] >;
C18×C3⋊C8 in GAP, Magma, Sage, TeX
C_{18}\times C_3\rtimes C_8 % in TeX
G:=Group("C18xC3:C8"); // GroupNames label
G:=SmallGroup(432,126);
// by ID
G=gap.SmallGroup(432,126);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,84,142,192,14118]);
// Polycyclic
G:=Group<a,b,c|a^18=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations