direct product, non-abelian, soluble
Aliases: C2×Q8.C18, C4○D4⋊2C18, (C2×C12).3A4, Q8⋊C9⋊3C22, (C6×Q8).6C6, C12.17(C2×A4), C6.3(C4.A4), (C2×Q8).2C18, Q8.1(C2×C18), C6.16(C22×A4), (C2×C4○D4)⋊C9, C3.(C2×C4.A4), (C2×Q8⋊C9)⋊4C2, (C6×C4○D4).C3, C4.6(C2×C3.A4), (C2×C6).25(C2×A4), (C3×C4○D4).7C6, (C2×C4).2(C3.A4), (C3×Q8).12(C2×C6), C22.9(C2×C3.A4), C2.5(C22×C3.A4), SmallGroup(288,347)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C2×Q8.C18 |
Generators and relations for C2×Q8.C18
G = < a,b,c,d | a2=b4=1, c2=d18=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=bc, dcd-1=b >
Subgroups: 225 in 85 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C9, C12, C12, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C18, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, C2×C4○D4, C36, C2×C18, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, Q8⋊C9, C2×C36, C6×C4○D4, C2×Q8⋊C9, Q8.C18, C2×Q8.C18
Quotients: C1, C2, C3, C22, C6, C9, A4, C2×C6, C18, C2×A4, C3.A4, C2×C18, C4.A4, C22×A4, C2×C3.A4, C2×C4.A4, Q8.C18, C22×C3.A4, C2×Q8.C18
►On 144 points
Generators in S144
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 109)(98 110)(99 111)(100 112)(101 113)(102 114)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)
(1 138 19 120)(2 56 20 38)(3 83 21 101)(4 141 22 123)(5 59 23 41)(6 86 24 104)(7 144 25 126)(8 62 26 44)(9 89 27 107)(10 111 28 129)(11 65 29 47)(12 92 30 74)(13 114 31 132)(14 68 32 50)(15 95 33 77)(16 117 34 135)(17 71 35 53)(18 98 36 80)(37 99 55 81)(39 140 57 122)(40 102 58 84)(42 143 60 125)(43 105 61 87)(45 110 63 128)(46 108 64 90)(48 113 66 131)(49 75 67 93)(51 116 69 134)(52 78 70 96)(54 119 72 137)(73 130 91 112)(76 133 94 115)(79 136 97 118)(82 139 100 121)(85 142 103 124)(88 109 106 127)
(1 81 19 99)(2 139 20 121)(3 57 21 39)(4 84 22 102)(5 142 23 124)(6 60 24 42)(7 87 25 105)(8 109 26 127)(9 63 27 45)(10 90 28 108)(11 112 29 130)(12 66 30 48)(13 93 31 75)(14 115 32 133)(15 69 33 51)(16 96 34 78)(17 118 35 136)(18 72 36 54)(37 138 55 120)(38 100 56 82)(40 141 58 123)(41 103 59 85)(43 144 61 126)(44 106 62 88)(46 111 64 129)(47 73 65 91)(49 114 67 132)(50 76 68 94)(52 117 70 135)(53 79 71 97)(74 131 92 113)(77 134 95 116)(80 137 98 119)(83 140 101 122)(86 143 104 125)(89 110 107 128)
(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)
G:=sub<Sym(144)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,109)(98,110)(99,111)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120), (1,138,19,120)(2,56,20,38)(3,83,21,101)(4,141,22,123)(5,59,23,41)(6,86,24,104)(7,144,25,126)(8,62,26,44)(9,89,27,107)(10,111,28,129)(11,65,29,47)(12,92,30,74)(13,114,31,132)(14,68,32,50)(15,95,33,77)(16,117,34,135)(17,71,35,53)(18,98,36,80)(37,99,55,81)(39,140,57,122)(40,102,58,84)(42,143,60,125)(43,105,61,87)(45,110,63,128)(46,108,64,90)(48,113,66,131)(49,75,67,93)(51,116,69,134)(52,78,70,96)(54,119,72,137)(73,130,91,112)(76,133,94,115)(79,136,97,118)(82,139,100,121)(85,142,103,124)(88,109,106,127), (1,81,19,99)(2,139,20,121)(3,57,21,39)(4,84,22,102)(5,142,23,124)(6,60,24,42)(7,87,25,105)(8,109,26,127)(9,63,27,45)(10,90,28,108)(11,112,29,130)(12,66,30,48)(13,93,31,75)(14,115,32,133)(15,69,33,51)(16,96,34,78)(17,118,35,136)(18,72,36,54)(37,138,55,120)(38,100,56,82)(40,141,58,123)(41,103,59,85)(43,144,61,126)(44,106,62,88)(46,111,64,129)(47,73,65,91)(49,114,67,132)(50,76,68,94)(52,117,70,135)(53,79,71,97)(74,131,92,113)(77,134,95,116)(80,137,98,119)(83,140,101,122)(86,143,104,125)(89,110,107,128), (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)>;
G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,109)(98,110)(99,111)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120), (1,138,19,120)(2,56,20,38)(3,83,21,101)(4,141,22,123)(5,59,23,41)(6,86,24,104)(7,144,25,126)(8,62,26,44)(9,89,27,107)(10,111,28,129)(11,65,29,47)(12,92,30,74)(13,114,31,132)(14,68,32,50)(15,95,33,77)(16,117,34,135)(17,71,35,53)(18,98,36,80)(37,99,55,81)(39,140,57,122)(40,102,58,84)(42,143,60,125)(43,105,61,87)(45,110,63,128)(46,108,64,90)(48,113,66,131)(49,75,67,93)(51,116,69,134)(52,78,70,96)(54,119,72,137)(73,130,91,112)(76,133,94,115)(79,136,97,118)(82,139,100,121)(85,142,103,124)(88,109,106,127), (1,81,19,99)(2,139,20,121)(3,57,21,39)(4,84,22,102)(5,142,23,124)(6,60,24,42)(7,87,25,105)(8,109,26,127)(9,63,27,45)(10,90,28,108)(11,112,29,130)(12,66,30,48)(13,93,31,75)(14,115,32,133)(15,69,33,51)(16,96,34,78)(17,118,35,136)(18,72,36,54)(37,138,55,120)(38,100,56,82)(40,141,58,123)(41,103,59,85)(43,144,61,126)(44,106,62,88)(46,111,64,129)(47,73,65,91)(49,114,67,132)(50,76,68,94)(52,117,70,135)(53,79,71,97)(74,131,92,113)(77,134,95,116)(80,137,98,119)(83,140,101,122)(86,143,104,125)(89,110,107,128), (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) );
G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,141),(94,142),(95,143),(96,144),(97,109),(98,110),(99,111),(100,112),(101,113),(102,114),(103,115),(104,116),(105,117),(106,118),(107,119),(108,120)], [(1,138,19,120),(2,56,20,38),(3,83,21,101),(4,141,22,123),(5,59,23,41),(6,86,24,104),(7,144,25,126),(8,62,26,44),(9,89,27,107),(10,111,28,129),(11,65,29,47),(12,92,30,74),(13,114,31,132),(14,68,32,50),(15,95,33,77),(16,117,34,135),(17,71,35,53),(18,98,36,80),(37,99,55,81),(39,140,57,122),(40,102,58,84),(42,143,60,125),(43,105,61,87),(45,110,63,128),(46,108,64,90),(48,113,66,131),(49,75,67,93),(51,116,69,134),(52,78,70,96),(54,119,72,137),(73,130,91,112),(76,133,94,115),(79,136,97,118),(82,139,100,121),(85,142,103,124),(88,109,106,127)], [(1,81,19,99),(2,139,20,121),(3,57,21,39),(4,84,22,102),(5,142,23,124),(6,60,24,42),(7,87,25,105),(8,109,26,127),(9,63,27,45),(10,90,28,108),(11,112,29,130),(12,66,30,48),(13,93,31,75),(14,115,32,133),(15,69,33,51),(16,96,34,78),(17,118,35,136),(18,72,36,54),(37,138,55,120),(38,100,56,82),(40,141,58,123),(41,103,59,85),(43,144,61,126),(44,106,62,88),(46,111,64,129),(47,73,65,91),(49,114,67,132),(50,76,68,94),(52,117,70,135),(53,79,71,97),(74,131,92,113),(77,134,95,116),(80,137,98,119),(83,140,101,122),(86,143,104,125),(89,110,107,128)], [(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)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 9A | ··· | 9F | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 18A | ··· | 18R | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 4 | ··· | 4 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | C4.A4 | Q8.C18 | A4 | C2×A4 | C2×A4 | C3.A4 | C2×C3.A4 | C2×C3.A4 |
| kernel | C2×Q8.C18 | C2×Q8⋊C9 | Q8.C18 | C6×C4○D4 | C6×Q8 | C3×C4○D4 | C2×C4○D4 | C2×Q8 | C4○D4 | C6 | C2 | C2×C12 | C12 | C2×C6 | C2×C4 | C4 | C22 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 24 | 1 | 2 | 1 | 2 | 4 | 2 |
Matrix representation of C2×Q8.C18 ►in GL3(𝔽37) generated by
| 36 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 1 | 0 | 0 |
| 0 | 1 | 36 |
| 0 | 2 | 36 |
| 1 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 12 | 31 |
| 26 | 0 | 0 |
| 0 | 25 | 7 |
| 0 | 23 | 14 |
G:=sub<GL(3,GF(37))| [36,0,0,0,36,0,0,0,36],[1,0,0,0,1,2,0,36,36],[1,0,0,0,6,12,0,0,31],[26,0,0,0,25,23,0,7,14] >;
C2×Q8.C18 in GAP, Magma, Sage, TeX
C_2\times Q_8.C_{18} % in TeX
G:=Group("C2xQ8.C18"); // GroupNames label
G:=SmallGroup(288,347);
// by ID
G=gap.SmallGroup(288,347);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1016,79,648,172,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=1,c^2=d^18=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b*c,d*c*d^-1=b>;
// generators/relations