direct product, metabelian, supersoluble, monomial
Aliases: C18×Dic6, C36.74D6, C6⋊(Q8×C9), (C3×C18)⋊2Q8, C3⋊1(Q8×C18), (C6×Dic6).C3, C4.11(S3×C18), (C6×C12).35C6, (C2×C36).20S3, (C6×C36).20C2, (C2×C12).4C18, (C2×C18).51D6, C3.4(C6×Dic6), C12.109(S3×C6), C12.11(C2×C18), C32.2(C6×Q8), C6.1(C22×C18), C22.8(S3×C18), C62.53(C2×C6), C6.16(C3×Dic6), C18.49(C22×S3), (C6×C18).26C22, (C3×C36).79C22, (C3×C18).28C23, (C2×Dic3).3C18, Dic3.1(C2×C18), (C6×Dic3).10C6, (C3×Dic6).11C6, (Dic3×C18).8C2, (C9×Dic3).13C22, (C3×C9)⋊5(C2×Q8), C2.3(S3×C2×C18), C6.62(S3×C2×C6), (C2×C4).4(S3×C9), (C2×C6).85(S3×C6), (C3×C6).11(C3×Q8), (C2×C12).42(C3×S3), (C2×C6).11(C2×C18), (C3×C12).80(C2×C6), (C3×C6).38(C22×C6), (C3×Dic3).10(C2×C6), SmallGroup(432,341)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C18×Dic6
G = < a,b,c | a18=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 212 in 130 conjugacy classes, 81 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×Q8, C18, C18, C18, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×C9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C2×Dic6, C6×Q8, C3×C18, C3×C18, C2×C36, C2×C36, Q8×C9, C3×Dic6, C6×Dic3, C6×C12, C9×Dic3, C3×C36, C6×C18, Q8×C18, C6×Dic6, C9×Dic6, Dic3×C18, C6×C36, C18×Dic6
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, C9, D6, C2×C6, C2×Q8, C18, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, C2×C18, S3×C6, C2×Dic6, C6×Q8, S3×C9, Q8×C9, C22×C18, C3×Dic6, S3×C2×C6, S3×C18, Q8×C18, C6×Dic6, C9×Dic6, S3×C2×C18, C18×Dic6
►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)
(1 136 106 120 13 130 100 114 7 142 94 126)(2 137 107 121 14 131 101 115 8 143 95 109)(3 138 108 122 15 132 102 116 9 144 96 110)(4 139 91 123 16 133 103 117 10 127 97 111)(5 140 92 124 17 134 104 118 11 128 98 112)(6 141 93 125 18 135 105 119 12 129 99 113)(19 79 64 52 25 85 70 40 31 73 58 46)(20 80 65 53 26 86 71 41 32 74 59 47)(21 81 66 54 27 87 72 42 33 75 60 48)(22 82 67 37 28 88 55 43 34 76 61 49)(23 83 68 38 29 89 56 44 35 77 62 50)(24 84 69 39 30 90 57 45 36 78 63 51)
(1 66 100 33)(2 67 101 34)(3 68 102 35)(4 69 103 36)(5 70 104 19)(6 71 105 20)(7 72 106 21)(8 55 107 22)(9 56 108 23)(10 57 91 24)(11 58 92 25)(12 59 93 26)(13 60 94 27)(14 61 95 28)(15 62 96 29)(16 63 97 30)(17 64 98 31)(18 65 99 32)(37 131 76 109)(38 132 77 110)(39 133 78 111)(40 134 79 112)(41 135 80 113)(42 136 81 114)(43 137 82 115)(44 138 83 116)(45 139 84 117)(46 140 85 118)(47 141 86 119)(48 142 87 120)(49 143 88 121)(50 144 89 122)(51 127 90 123)(52 128 73 124)(53 129 74 125)(54 130 75 126)
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), (1,136,106,120,13,130,100,114,7,142,94,126)(2,137,107,121,14,131,101,115,8,143,95,109)(3,138,108,122,15,132,102,116,9,144,96,110)(4,139,91,123,16,133,103,117,10,127,97,111)(5,140,92,124,17,134,104,118,11,128,98,112)(6,141,93,125,18,135,105,119,12,129,99,113)(19,79,64,52,25,85,70,40,31,73,58,46)(20,80,65,53,26,86,71,41,32,74,59,47)(21,81,66,54,27,87,72,42,33,75,60,48)(22,82,67,37,28,88,55,43,34,76,61,49)(23,83,68,38,29,89,56,44,35,77,62,50)(24,84,69,39,30,90,57,45,36,78,63,51), (1,66,100,33)(2,67,101,34)(3,68,102,35)(4,69,103,36)(5,70,104,19)(6,71,105,20)(7,72,106,21)(8,55,107,22)(9,56,108,23)(10,57,91,24)(11,58,92,25)(12,59,93,26)(13,60,94,27)(14,61,95,28)(15,62,96,29)(16,63,97,30)(17,64,98,31)(18,65,99,32)(37,131,76,109)(38,132,77,110)(39,133,78,111)(40,134,79,112)(41,135,80,113)(42,136,81,114)(43,137,82,115)(44,138,83,116)(45,139,84,117)(46,140,85,118)(47,141,86,119)(48,142,87,120)(49,143,88,121)(50,144,89,122)(51,127,90,123)(52,128,73,124)(53,129,74,125)(54,130,75,126)>;
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), (1,136,106,120,13,130,100,114,7,142,94,126)(2,137,107,121,14,131,101,115,8,143,95,109)(3,138,108,122,15,132,102,116,9,144,96,110)(4,139,91,123,16,133,103,117,10,127,97,111)(5,140,92,124,17,134,104,118,11,128,98,112)(6,141,93,125,18,135,105,119,12,129,99,113)(19,79,64,52,25,85,70,40,31,73,58,46)(20,80,65,53,26,86,71,41,32,74,59,47)(21,81,66,54,27,87,72,42,33,75,60,48)(22,82,67,37,28,88,55,43,34,76,61,49)(23,83,68,38,29,89,56,44,35,77,62,50)(24,84,69,39,30,90,57,45,36,78,63,51), (1,66,100,33)(2,67,101,34)(3,68,102,35)(4,69,103,36)(5,70,104,19)(6,71,105,20)(7,72,106,21)(8,55,107,22)(9,56,108,23)(10,57,91,24)(11,58,92,25)(12,59,93,26)(13,60,94,27)(14,61,95,28)(15,62,96,29)(16,63,97,30)(17,64,98,31)(18,65,99,32)(37,131,76,109)(38,132,77,110)(39,133,78,111)(40,134,79,112)(41,135,80,113)(42,136,81,114)(43,137,82,115)(44,138,83,116)(45,139,84,117)(46,140,85,118)(47,141,86,119)(48,142,87,120)(49,143,88,121)(50,144,89,122)(51,127,90,123)(52,128,73,124)(53,129,74,125)(54,130,75,126) );
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)], [(1,136,106,120,13,130,100,114,7,142,94,126),(2,137,107,121,14,131,101,115,8,143,95,109),(3,138,108,122,15,132,102,116,9,144,96,110),(4,139,91,123,16,133,103,117,10,127,97,111),(5,140,92,124,17,134,104,118,11,128,98,112),(6,141,93,125,18,135,105,119,12,129,99,113),(19,79,64,52,25,85,70,40,31,73,58,46),(20,80,65,53,26,86,71,41,32,74,59,47),(21,81,66,54,27,87,72,42,33,75,60,48),(22,82,67,37,28,88,55,43,34,76,61,49),(23,83,68,38,29,89,56,44,35,77,62,50),(24,84,69,39,30,90,57,45,36,78,63,51)], [(1,66,100,33),(2,67,101,34),(3,68,102,35),(4,69,103,36),(5,70,104,19),(6,71,105,20),(7,72,106,21),(8,55,107,22),(9,56,108,23),(10,57,91,24),(11,58,92,25),(12,59,93,26),(13,60,94,27),(14,61,95,28),(15,62,96,29),(16,63,97,30),(17,64,98,31),(18,65,99,32),(37,131,76,109),(38,132,77,110),(39,133,78,111),(40,134,79,112),(41,135,80,113),(42,136,81,114),(43,137,82,115),(44,138,83,116),(45,139,84,117),(46,140,85,118),(47,141,86,119),(48,142,87,120),(49,143,88,121),(50,144,89,122),(51,127,90,123),(52,128,73,124),(53,129,74,125),(54,130,75,126)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12P | 12Q | ··· | 12X | 18A | ··· | 18R | 18S | ··· | 18AJ | 36A | ··· | 36AJ | 36AK | ··· | 36BH |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 |
162 irreducible representations
| dim | 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 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | S3 | Q8 | D6 | D6 | C3×S3 | Dic6 | C3×Q8 | S3×C6 | S3×C6 | S3×C9 | Q8×C9 | C3×Dic6 | S3×C18 | S3×C18 | C9×Dic6 |
| kernel | C18×Dic6 | C9×Dic6 | Dic3×C18 | C6×C36 | C6×Dic6 | C3×Dic6 | C6×Dic3 | C6×C12 | C2×Dic6 | Dic6 | C2×Dic3 | C2×C12 | C2×C36 | C3×C18 | C36 | C2×C18 | C2×C12 | C18 | C3×C6 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 6 | 24 | 12 | 6 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 6 | 12 | 8 | 12 | 6 | 24 |
Matrix representation of C18×Dic6 ►in GL5(𝔽37)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 20 | 20 |
| 0 | 0 | 0 | 4 | 17 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 23 |
| 0 | 0 | 0 | 32 | 22 |
G:=sub<GL(5,GF(37))| [4,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[36,0,0,0,0,0,11,0,0,0,0,0,27,0,0,0,0,0,20,4,0,0,0,20,17],[36,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,15,32,0,0,0,23,22] >;
C18×Dic6 in GAP, Magma, Sage, TeX
C_{18}\times {\rm Dic}_6 % in TeX
G:=Group("C18xDic6"); // GroupNames label
G:=SmallGroup(432,341);
// by ID
G=gap.SmallGroup(432,341);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,590,142,192,14118]);
// Polycyclic
G:=Group<a,b,c|a^18=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations