direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C6×C3⋊C8, C12.14C62, C62.21C12, C62.27Dic3, C6⋊(C3×C24), (C3×C6)⋊5C24, C3⋊2(C6×C24), C6.6(C6×C12), (C32×C6)⋊5C8, C33⋊19(C2×C8), (C6×C12).42C6, (C6×C12).59S3, C12.3(C3×C12), (C3×C62).6C4, C12.123(S3×C6), (C3×C12).24C12, C32⋊11(C2×C24), (C3×C12).237D6, C6.35(C6×Dic3), (C32×C12).13C4, C12.26(C3×Dic3), (C3×C12).33Dic3, C4.3(C32×Dic3), (C32×C12).89C22, C22.2(C32×Dic3), C4.14(S3×C3×C6), (C3×C6×C12).13C2, C2.1(Dic3×C3×C6), (C2×C12).11(C3×C6), (C2×C12).51(C3×S3), (C3×C12).94(C2×C6), (C3×C6).56(C2×C12), (C2×C6).13(C3×C12), (C2×C4).5(S3×C32), (C32×C6).64(C2×C4), (C3×C6).76(C2×Dic3), (C2×C6).28(C3×Dic3), SmallGroup(432,469)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C3×C6×C3⋊C8 |
Generators and relations for C3×C6×C3⋊C8
G = < a,b,c,d | a3=b6=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 296 in 196 conjugacy classes, 114 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C32, C32, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C33, C3×C12, C3×C12, C3×C12, C62, C62, C62, C2×C3⋊C8, C2×C24, C32×C6, C32×C6, C3×C3⋊C8, C3×C24, C6×C12, C6×C12, C6×C12, C32×C12, C3×C62, C6×C3⋊C8, C6×C24, C32×C3⋊C8, C3×C6×C12, C3×C6×C3⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C32, Dic3, C12, D6, C2×C6, C2×C8, C3×S3, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C3×Dic3, C3×C12, S3×C6, C62, C2×C3⋊C8, C2×C24, S3×C32, C3×C3⋊C8, C3×C24, C6×Dic3, C6×C12, C32×Dic3, S3×C3×C6, C6×C3⋊C8, C6×C24, C32×C3⋊C8, Dic3×C3×C6, C3×C6×C3⋊C8
►On 144 points
Generators in S144
(1 81 78)(2 82 79)(3 83 80)(4 84 73)(5 85 74)(6 86 75)(7 87 76)(8 88 77)(9 139 110)(10 140 111)(11 141 112)(12 142 105)(13 143 106)(14 144 107)(15 137 108)(16 138 109)(17 104 127)(18 97 128)(19 98 121)(20 99 122)(21 100 123)(22 101 124)(23 102 125)(24 103 126)(25 129 50)(26 130 51)(27 131 52)(28 132 53)(29 133 54)(30 134 55)(31 135 56)(32 136 49)(33 119 42)(34 120 43)(35 113 44)(36 114 45)(37 115 46)(38 116 47)(39 117 48)(40 118 41)(57 95 65)(58 96 66)(59 89 67)(60 90 68)(61 91 69)(62 92 70)(63 93 71)(64 94 72)
(1 94 39 55 143 23)(2 95 40 56 144 24)(3 96 33 49 137 17)(4 89 34 50 138 18)(5 90 35 51 139 19)(6 91 36 52 140 20)(7 92 37 53 141 21)(8 93 38 54 142 22)(9 121 74 60 44 130)(10 122 75 61 45 131)(11 123 76 62 46 132)(12 124 77 63 47 133)(13 125 78 64 48 134)(14 126 79 57 41 135)(15 127 80 58 42 136)(16 128 73 59 43 129)(25 109 97 84 67 120)(26 110 98 85 68 113)(27 111 99 86 69 114)(28 112 100 87 70 115)(29 105 101 88 71 116)(30 106 102 81 72 117)(31 107 103 82 65 118)(32 108 104 83 66 119)
(1 39 143)(2 144 40)(3 33 137)(4 138 34)(5 35 139)(6 140 36)(7 37 141)(8 142 38)(9 74 44)(10 45 75)(11 76 46)(12 47 77)(13 78 48)(14 41 79)(15 80 42)(16 43 73)(17 96 49)(18 50 89)(19 90 51)(20 52 91)(21 92 53)(22 54 93)(23 94 55)(24 56 95)(25 67 97)(26 98 68)(27 69 99)(28 100 70)(29 71 101)(30 102 72)(31 65 103)(32 104 66)(57 126 135)(58 136 127)(59 128 129)(60 130 121)(61 122 131)(62 132 123)(63 124 133)(64 134 125)(81 117 106)(82 107 118)(83 119 108)(84 109 120)(85 113 110)(86 111 114)(87 115 112)(88 105 116)
(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)| (1,81,78)(2,82,79)(3,83,80)(4,84,73)(5,85,74)(6,86,75)(7,87,76)(8,88,77)(9,139,110)(10,140,111)(11,141,112)(12,142,105)(13,143,106)(14,144,107)(15,137,108)(16,138,109)(17,104,127)(18,97,128)(19,98,121)(20,99,122)(21,100,123)(22,101,124)(23,102,125)(24,103,126)(25,129,50)(26,130,51)(27,131,52)(28,132,53)(29,133,54)(30,134,55)(31,135,56)(32,136,49)(33,119,42)(34,120,43)(35,113,44)(36,114,45)(37,115,46)(38,116,47)(39,117,48)(40,118,41)(57,95,65)(58,96,66)(59,89,67)(60,90,68)(61,91,69)(62,92,70)(63,93,71)(64,94,72), (1,94,39,55,143,23)(2,95,40,56,144,24)(3,96,33,49,137,17)(4,89,34,50,138,18)(5,90,35,51,139,19)(6,91,36,52,140,20)(7,92,37,53,141,21)(8,93,38,54,142,22)(9,121,74,60,44,130)(10,122,75,61,45,131)(11,123,76,62,46,132)(12,124,77,63,47,133)(13,125,78,64,48,134)(14,126,79,57,41,135)(15,127,80,58,42,136)(16,128,73,59,43,129)(25,109,97,84,67,120)(26,110,98,85,68,113)(27,111,99,86,69,114)(28,112,100,87,70,115)(29,105,101,88,71,116)(30,106,102,81,72,117)(31,107,103,82,65,118)(32,108,104,83,66,119), (1,39,143)(2,144,40)(3,33,137)(4,138,34)(5,35,139)(6,140,36)(7,37,141)(8,142,38)(9,74,44)(10,45,75)(11,76,46)(12,47,77)(13,78,48)(14,41,79)(15,80,42)(16,43,73)(17,96,49)(18,50,89)(19,90,51)(20,52,91)(21,92,53)(22,54,93)(23,94,55)(24,56,95)(25,67,97)(26,98,68)(27,69,99)(28,100,70)(29,71,101)(30,102,72)(31,65,103)(32,104,66)(57,126,135)(58,136,127)(59,128,129)(60,130,121)(61,122,131)(62,132,123)(63,124,133)(64,134,125)(81,117,106)(82,107,118)(83,119,108)(84,109,120)(85,113,110)(86,111,114)(87,115,112)(88,105,116), (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( (1,81,78)(2,82,79)(3,83,80)(4,84,73)(5,85,74)(6,86,75)(7,87,76)(8,88,77)(9,139,110)(10,140,111)(11,141,112)(12,142,105)(13,143,106)(14,144,107)(15,137,108)(16,138,109)(17,104,127)(18,97,128)(19,98,121)(20,99,122)(21,100,123)(22,101,124)(23,102,125)(24,103,126)(25,129,50)(26,130,51)(27,131,52)(28,132,53)(29,133,54)(30,134,55)(31,135,56)(32,136,49)(33,119,42)(34,120,43)(35,113,44)(36,114,45)(37,115,46)(38,116,47)(39,117,48)(40,118,41)(57,95,65)(58,96,66)(59,89,67)(60,90,68)(61,91,69)(62,92,70)(63,93,71)(64,94,72), (1,94,39,55,143,23)(2,95,40,56,144,24)(3,96,33,49,137,17)(4,89,34,50,138,18)(5,90,35,51,139,19)(6,91,36,52,140,20)(7,92,37,53,141,21)(8,93,38,54,142,22)(9,121,74,60,44,130)(10,122,75,61,45,131)(11,123,76,62,46,132)(12,124,77,63,47,133)(13,125,78,64,48,134)(14,126,79,57,41,135)(15,127,80,58,42,136)(16,128,73,59,43,129)(25,109,97,84,67,120)(26,110,98,85,68,113)(27,111,99,86,69,114)(28,112,100,87,70,115)(29,105,101,88,71,116)(30,106,102,81,72,117)(31,107,103,82,65,118)(32,108,104,83,66,119), (1,39,143)(2,144,40)(3,33,137)(4,138,34)(5,35,139)(6,140,36)(7,37,141)(8,142,38)(9,74,44)(10,45,75)(11,76,46)(12,47,77)(13,78,48)(14,41,79)(15,80,42)(16,43,73)(17,96,49)(18,50,89)(19,90,51)(20,52,91)(21,92,53)(22,54,93)(23,94,55)(24,56,95)(25,67,97)(26,98,68)(27,69,99)(28,100,70)(29,71,101)(30,102,72)(31,65,103)(32,104,66)(57,126,135)(58,136,127)(59,128,129)(60,130,121)(61,122,131)(62,132,123)(63,124,133)(64,134,125)(81,117,106)(82,107,118)(83,119,108)(84,109,120)(85,113,110)(86,111,114)(87,115,112)(88,105,116), (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([[(1,81,78),(2,82,79),(3,83,80),(4,84,73),(5,85,74),(6,86,75),(7,87,76),(8,88,77),(9,139,110),(10,140,111),(11,141,112),(12,142,105),(13,143,106),(14,144,107),(15,137,108),(16,138,109),(17,104,127),(18,97,128),(19,98,121),(20,99,122),(21,100,123),(22,101,124),(23,102,125),(24,103,126),(25,129,50),(26,130,51),(27,131,52),(28,132,53),(29,133,54),(30,134,55),(31,135,56),(32,136,49),(33,119,42),(34,120,43),(35,113,44),(36,114,45),(37,115,46),(38,116,47),(39,117,48),(40,118,41),(57,95,65),(58,96,66),(59,89,67),(60,90,68),(61,91,69),(62,92,70),(63,93,71),(64,94,72)], [(1,94,39,55,143,23),(2,95,40,56,144,24),(3,96,33,49,137,17),(4,89,34,50,138,18),(5,90,35,51,139,19),(6,91,36,52,140,20),(7,92,37,53,141,21),(8,93,38,54,142,22),(9,121,74,60,44,130),(10,122,75,61,45,131),(11,123,76,62,46,132),(12,124,77,63,47,133),(13,125,78,64,48,134),(14,126,79,57,41,135),(15,127,80,58,42,136),(16,128,73,59,43,129),(25,109,97,84,67,120),(26,110,98,85,68,113),(27,111,99,86,69,114),(28,112,100,87,70,115),(29,105,101,88,71,116),(30,106,102,81,72,117),(31,107,103,82,65,118),(32,108,104,83,66,119)], [(1,39,143),(2,144,40),(3,33,137),(4,138,34),(5,35,139),(6,140,36),(7,37,141),(8,142,38),(9,74,44),(10,45,75),(11,76,46),(12,47,77),(13,78,48),(14,41,79),(15,80,42),(16,43,73),(17,96,49),(18,50,89),(19,90,51),(20,52,91),(21,92,53),(22,54,93),(23,94,55),(24,56,95),(25,67,97),(26,98,68),(27,69,99),(28,100,70),(29,71,101),(30,102,72),(31,65,103),(32,104,66),(57,126,135),(58,136,127),(59,128,129),(60,130,121),(61,122,131),(62,132,123),(63,124,133),(64,134,125),(81,117,106),(82,107,118),(83,119,108),(84,109,120),(85,113,110),(86,111,114),(87,115,112),(88,105,116)], [(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)]])
216 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 4D | 6A | ··· | 6X | 6Y | ··· | 6AY | 8A | ··· | 8H | 12A | ··· | 12AF | 12AG | ··· | 12BP | 24A | ··· | 24BL |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | Dic3 | D6 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | S3×C6 | C3×Dic3 | C3×C3⋊C8 |
| kernel | C3×C6×C3⋊C8 | C32×C3⋊C8 | C3×C6×C12 | C6×C3⋊C8 | C32×C12 | C3×C62 | C3×C3⋊C8 | C6×C12 | C32×C6 | C3×C12 | C62 | C3×C6 | C6×C12 | C3×C12 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 16 | 8 | 8 | 16 | 16 | 64 | 1 | 1 | 1 | 1 | 8 | 4 | 8 | 8 | 8 | 32 |
Matrix representation of C3×C6×C3⋊C8 ►in GL3(𝔽73) generated by
| 8 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 65 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 64 |
| 72 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 46 | 0 |
G:=sub<GL(3,GF(73))| [8,0,0,0,1,0,0,0,1],[65,0,0,0,64,0,0,0,64],[1,0,0,0,8,0,0,0,64],[72,0,0,0,0,46,0,1,0] >;
C3×C6×C3⋊C8 in GAP, Magma, Sage, TeX
C_3\times C_6\times C_3\rtimes C_8
% in TeX
G:=Group("C3xC6xC3:C8"); // GroupNames label
G:=SmallGroup(432,469);
// by ID
G=gap.SmallGroup(432,469);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,102,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations