direct product, metabelian, nilpotent (class 3), monomial
Aliases: D8×C3×C6, C8⋊2C62, D4⋊1C62, C62.144D4, (C2×C24)⋊8C6, (C6×C24)⋊13C2, C24⋊10(C2×C6), (C6×D4)⋊13C6, C6.91(C6×D4), C12.50(C3×D4), C4.1(C2×C62), C4.6(D4×C32), (C3×C24)⋊28C22, (C3×C12).147D4, (C2×C4).25C62, C12.55(C22×C6), (C3×C12).185C23, (C6×C12).374C22, (D4×C32)⋊28C22, C22.14(D4×C32), (C2×C8)⋊3(C3×C6), (D4×C3×C6)⋊22C2, C2.11(D4×C3×C6), (C2×D4)⋊4(C3×C6), (C3×D4)⋊10(C2×C6), (C2×C6).72(C3×D4), (C3×C6).308(C2×D4), (C2×C12).161(C2×C6), SmallGroup(288,829)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C3×C6
G = < a,b,c,d | a3=b6=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 420 in 228 conjugacy classes, 132 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, D4, D4, C23, C32, C12, C2×C6, C2×C6, C2×C8, D8, C2×D4, C3×C6, C3×C6, C3×C6, C24, C2×C12, C3×D4, C3×D4, C22×C6, C2×D8, C3×C12, C62, C62, C2×C24, C3×D8, C6×D4, C3×C24, C6×C12, D4×C32, D4×C32, C2×C62, C6×D8, C6×C24, C32×D8, D4×C3×C6, D8×C3×C6
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, D8, C2×D4, C3×C6, C3×D4, C22×C6, C2×D8, C62, C3×D8, C6×D4, D4×C32, C2×C62, C6×D8, C32×D8, D4×C3×C6, D8×C3×C6
►On 144 points
Generators in S144
(1 32 24)(2 25 17)(3 26 18)(4 27 19)(5 28 20)(6 29 21)(7 30 22)(8 31 23)(9 35 130)(10 36 131)(11 37 132)(12 38 133)(13 39 134)(14 40 135)(15 33 136)(16 34 129)(41 92 49)(42 93 50)(43 94 51)(44 95 52)(45 96 53)(46 89 54)(47 90 55)(48 91 56)(57 78 65)(58 79 66)(59 80 67)(60 73 68)(61 74 69)(62 75 70)(63 76 71)(64 77 72)(81 127 119)(82 128 120)(83 121 113)(84 122 114)(85 123 115)(86 124 116)(87 125 117)(88 126 118)(97 143 105)(98 144 106)(99 137 107)(100 138 108)(101 139 109)(102 140 110)(103 141 111)(104 142 112)
(1 101 81 58 46 131)(2 102 82 59 47 132)(3 103 83 60 48 133)(4 104 84 61 41 134)(5 97 85 62 42 135)(6 98 86 63 43 136)(7 99 87 64 44 129)(8 100 88 57 45 130)(9 31 138 126 78 96)(10 32 139 127 79 89)(11 25 140 128 80 90)(12 26 141 121 73 91)(13 27 142 122 74 92)(14 28 143 123 75 93)(15 29 144 124 76 94)(16 30 137 125 77 95)(17 110 120 67 55 37)(18 111 113 68 56 38)(19 112 114 69 49 39)(20 105 115 70 50 40)(21 106 116 71 51 33)(22 107 117 72 52 34)(23 108 118 65 53 35)(24 109 119 66 54 36)
(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 61)(2 60)(3 59)(4 58)(5 57)(6 64)(7 63)(8 62)(9 123)(10 122)(11 121)(12 128)(13 127)(14 126)(15 125)(16 124)(17 68)(18 67)(19 66)(20 65)(21 72)(22 71)(23 70)(24 69)(25 73)(26 80)(27 79)(28 78)(29 77)(30 76)(31 75)(32 74)(33 117)(34 116)(35 115)(36 114)(37 113)(38 120)(39 119)(40 118)(41 101)(42 100)(43 99)(44 98)(45 97)(46 104)(47 103)(48 102)(49 109)(50 108)(51 107)(52 106)(53 105)(54 112)(55 111)(56 110)(81 134)(82 133)(83 132)(84 131)(85 130)(86 129)(87 136)(88 135)(89 142)(90 141)(91 140)(92 139)(93 138)(94 137)(95 144)(96 143)
G:=sub<Sym(144)| (1,32,24)(2,25,17)(3,26,18)(4,27,19)(5,28,20)(6,29,21)(7,30,22)(8,31,23)(9,35,130)(10,36,131)(11,37,132)(12,38,133)(13,39,134)(14,40,135)(15,33,136)(16,34,129)(41,92,49)(42,93,50)(43,94,51)(44,95,52)(45,96,53)(46,89,54)(47,90,55)(48,91,56)(57,78,65)(58,79,66)(59,80,67)(60,73,68)(61,74,69)(62,75,70)(63,76,71)(64,77,72)(81,127,119)(82,128,120)(83,121,113)(84,122,114)(85,123,115)(86,124,116)(87,125,117)(88,126,118)(97,143,105)(98,144,106)(99,137,107)(100,138,108)(101,139,109)(102,140,110)(103,141,111)(104,142,112), (1,101,81,58,46,131)(2,102,82,59,47,132)(3,103,83,60,48,133)(4,104,84,61,41,134)(5,97,85,62,42,135)(6,98,86,63,43,136)(7,99,87,64,44,129)(8,100,88,57,45,130)(9,31,138,126,78,96)(10,32,139,127,79,89)(11,25,140,128,80,90)(12,26,141,121,73,91)(13,27,142,122,74,92)(14,28,143,123,75,93)(15,29,144,124,76,94)(16,30,137,125,77,95)(17,110,120,67,55,37)(18,111,113,68,56,38)(19,112,114,69,49,39)(20,105,115,70,50,40)(21,106,116,71,51,33)(22,107,117,72,52,34)(23,108,118,65,53,35)(24,109,119,66,54,36), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,64)(7,63)(8,62)(9,123)(10,122)(11,121)(12,128)(13,127)(14,126)(15,125)(16,124)(17,68)(18,67)(19,66)(20,65)(21,72)(22,71)(23,70)(24,69)(25,73)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,117)(34,116)(35,115)(36,114)(37,113)(38,120)(39,119)(40,118)(41,101)(42,100)(43,99)(44,98)(45,97)(46,104)(47,103)(48,102)(49,109)(50,108)(51,107)(52,106)(53,105)(54,112)(55,111)(56,110)(81,134)(82,133)(83,132)(84,131)(85,130)(86,129)(87,136)(88,135)(89,142)(90,141)(91,140)(92,139)(93,138)(94,137)(95,144)(96,143)>;
G:=Group( (1,32,24)(2,25,17)(3,26,18)(4,27,19)(5,28,20)(6,29,21)(7,30,22)(8,31,23)(9,35,130)(10,36,131)(11,37,132)(12,38,133)(13,39,134)(14,40,135)(15,33,136)(16,34,129)(41,92,49)(42,93,50)(43,94,51)(44,95,52)(45,96,53)(46,89,54)(47,90,55)(48,91,56)(57,78,65)(58,79,66)(59,80,67)(60,73,68)(61,74,69)(62,75,70)(63,76,71)(64,77,72)(81,127,119)(82,128,120)(83,121,113)(84,122,114)(85,123,115)(86,124,116)(87,125,117)(88,126,118)(97,143,105)(98,144,106)(99,137,107)(100,138,108)(101,139,109)(102,140,110)(103,141,111)(104,142,112), (1,101,81,58,46,131)(2,102,82,59,47,132)(3,103,83,60,48,133)(4,104,84,61,41,134)(5,97,85,62,42,135)(6,98,86,63,43,136)(7,99,87,64,44,129)(8,100,88,57,45,130)(9,31,138,126,78,96)(10,32,139,127,79,89)(11,25,140,128,80,90)(12,26,141,121,73,91)(13,27,142,122,74,92)(14,28,143,123,75,93)(15,29,144,124,76,94)(16,30,137,125,77,95)(17,110,120,67,55,37)(18,111,113,68,56,38)(19,112,114,69,49,39)(20,105,115,70,50,40)(21,106,116,71,51,33)(22,107,117,72,52,34)(23,108,118,65,53,35)(24,109,119,66,54,36), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,64)(7,63)(8,62)(9,123)(10,122)(11,121)(12,128)(13,127)(14,126)(15,125)(16,124)(17,68)(18,67)(19,66)(20,65)(21,72)(22,71)(23,70)(24,69)(25,73)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,117)(34,116)(35,115)(36,114)(37,113)(38,120)(39,119)(40,118)(41,101)(42,100)(43,99)(44,98)(45,97)(46,104)(47,103)(48,102)(49,109)(50,108)(51,107)(52,106)(53,105)(54,112)(55,111)(56,110)(81,134)(82,133)(83,132)(84,131)(85,130)(86,129)(87,136)(88,135)(89,142)(90,141)(91,140)(92,139)(93,138)(94,137)(95,144)(96,143) );
G=PermutationGroup([[(1,32,24),(2,25,17),(3,26,18),(4,27,19),(5,28,20),(6,29,21),(7,30,22),(8,31,23),(9,35,130),(10,36,131),(11,37,132),(12,38,133),(13,39,134),(14,40,135),(15,33,136),(16,34,129),(41,92,49),(42,93,50),(43,94,51),(44,95,52),(45,96,53),(46,89,54),(47,90,55),(48,91,56),(57,78,65),(58,79,66),(59,80,67),(60,73,68),(61,74,69),(62,75,70),(63,76,71),(64,77,72),(81,127,119),(82,128,120),(83,121,113),(84,122,114),(85,123,115),(86,124,116),(87,125,117),(88,126,118),(97,143,105),(98,144,106),(99,137,107),(100,138,108),(101,139,109),(102,140,110),(103,141,111),(104,142,112)], [(1,101,81,58,46,131),(2,102,82,59,47,132),(3,103,83,60,48,133),(4,104,84,61,41,134),(5,97,85,62,42,135),(6,98,86,63,43,136),(7,99,87,64,44,129),(8,100,88,57,45,130),(9,31,138,126,78,96),(10,32,139,127,79,89),(11,25,140,128,80,90),(12,26,141,121,73,91),(13,27,142,122,74,92),(14,28,143,123,75,93),(15,29,144,124,76,94),(16,30,137,125,77,95),(17,110,120,67,55,37),(18,111,113,68,56,38),(19,112,114,69,49,39),(20,105,115,70,50,40),(21,106,116,71,51,33),(22,107,117,72,52,34),(23,108,118,65,53,35),(24,109,119,66,54,36)], [(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)], [(1,61),(2,60),(3,59),(4,58),(5,57),(6,64),(7,63),(8,62),(9,123),(10,122),(11,121),(12,128),(13,127),(14,126),(15,125),(16,124),(17,68),(18,67),(19,66),(20,65),(21,72),(22,71),(23,70),(24,69),(25,73),(26,80),(27,79),(28,78),(29,77),(30,76),(31,75),(32,74),(33,117),(34,116),(35,115),(36,114),(37,113),(38,120),(39,119),(40,118),(41,101),(42,100),(43,99),(44,98),(45,97),(46,104),(47,103),(48,102),(49,109),(50,108),(51,107),(52,106),(53,105),(54,112),(55,111),(56,110),(81,134),(82,133),(83,132),(84,131),(85,130),(86,129),(87,136),(88,135),(89,142),(90,141),(91,140),(92,139),(93,138),(94,137),(95,144),(96,143)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | ··· | 3H | 4A | 4B | 6A | ··· | 6X | 6Y | ··· | 6BD | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | D8 | C3×D4 | C3×D4 | C3×D8 |
| kernel | D8×C3×C6 | C6×C24 | C32×D8 | D4×C3×C6 | C6×D8 | C2×C24 | C3×D8 | C6×D4 | C3×C12 | C62 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 1 | 4 | 2 | 8 | 8 | 32 | 16 | 1 | 1 | 4 | 8 | 8 | 32 |
Matrix representation of D8×C3×C6 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 8 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 57 | 57 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,72,0,0,0,0,9,0,0,0,0,9],[72,0,0,0,0,72,0,0,0,0,57,57,0,0,16,57],[1,0,0,0,0,72,0,0,0,0,57,16,0,0,16,16] >;
D8×C3×C6 in GAP, Magma, Sage, TeX
D_8\times C_3\times C_6
% in TeX
G:=Group("D8xC3xC6"); // GroupNames label
G:=SmallGroup(288,829);
// by ID
G=gap.SmallGroup(288,829);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1037,9077,4548,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations