direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C8×C3⋊S3, C24⋊26D6, C6⋊2(S3×C8), (C6×C24)⋊16C2, (C2×C24)⋊10S3, C12.82(C4×S3), C32⋊8(C22×C8), (C3×C24)⋊30C22, (C2×C12).425D6, C62.82(C2×C4), C12.206(C22×S3), (C6×C12).355C22, (C3×C12).175C23, C32⋊4C8⋊32C22, C3⋊3(S3×C2×C8), (C3×C6)⋊7(C2×C8), C6.72(S3×C2×C4), C4.23(C4×C3⋊S3), (C4×C3⋊S3).19C4, (C2×C6).52(C4×S3), C22.13(C4×C3⋊S3), C4.35(C22×C3⋊S3), (C3×C12).116(C2×C4), (C2×C32⋊4C8)⋊21C2, (C22×C3⋊S3).17C4, C3⋊Dic3.57(C2×C4), (C2×C3⋊Dic3).26C4, (C4×C3⋊S3).101C22, (C3×C6).103(C22×C4), C2.2(C2×C4×C3⋊S3), (C2×C4×C3⋊S3).32C2, (C2×C4).98(C2×C3⋊S3), (C2×C3⋊S3).52(C2×C4), SmallGroup(288,756)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C2×C8×C3⋊S3 |
Generators and relations for C2×C8×C3⋊S3
G = < a,b,c,d,e | a2=b8=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 660 in 228 conjugacy classes, 93 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, C22×C4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C32⋊4C8, C3×C24, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C2×C8, C8×C3⋊S3, C2×C32⋊4C8, C6×C24, C2×C4×C3⋊S3, C2×C8×C3⋊S3
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, C3⋊S3, C4×S3, C22×S3, C22×C8, C2×C3⋊S3, S3×C8, S3×C2×C4, C4×C3⋊S3, C22×C3⋊S3, S3×C2×C8, C8×C3⋊S3, C2×C4×C3⋊S3, C2×C8×C3⋊S3
►On 144 points
Generators in S144
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 97)(17 87)(18 88)(19 81)(20 82)(21 83)(22 84)(23 85)(24 86)(33 77)(34 78)(35 79)(36 80)(37 73)(38 74)(39 75)(40 76)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 129)(49 119)(50 120)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 127)(58 128)(59 121)(60 122)(61 123)(62 124)(63 125)(64 126)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(89 108)(90 109)(91 110)(92 111)(93 112)(94 105)(95 106)(96 107)
(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)
(1 104 130)(2 97 131)(3 98 132)(4 99 133)(5 100 134)(6 101 135)(7 102 136)(8 103 129)(9 43 29)(10 44 30)(11 45 31)(12 46 32)(13 47 25)(14 48 26)(15 41 27)(16 42 28)(17 56 39)(18 49 40)(19 50 33)(20 51 34)(21 52 35)(22 53 36)(23 54 37)(24 55 38)(57 110 142)(58 111 143)(59 112 144)(60 105 137)(61 106 138)(62 107 139)(63 108 140)(64 109 141)(65 122 94)(66 123 95)(67 124 96)(68 125 89)(69 126 90)(70 127 91)(71 128 92)(72 121 93)(73 85 116)(74 86 117)(75 87 118)(76 88 119)(77 81 120)(78 82 113)(79 83 114)(80 84 115)
(1 118 68)(2 119 69)(3 120 70)(4 113 71)(5 114 72)(6 115 65)(7 116 66)(8 117 67)(9 33 57)(10 34 58)(11 35 59)(12 36 60)(13 37 61)(14 38 62)(15 39 63)(16 40 64)(17 108 41)(18 109 42)(19 110 43)(20 111 44)(21 112 45)(22 105 46)(23 106 47)(24 107 48)(25 54 138)(26 55 139)(27 56 140)(28 49 141)(29 50 142)(30 51 143)(31 52 144)(32 53 137)(73 123 102)(74 124 103)(75 125 104)(76 126 97)(77 127 98)(78 128 99)(79 121 100)(80 122 101)(81 91 132)(82 92 133)(83 93 134)(84 94 135)(85 95 136)(86 96 129)(87 89 130)(88 90 131)
(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(33 110)(34 111)(35 112)(36 105)(37 106)(38 107)(39 108)(40 109)(49 141)(50 142)(51 143)(52 144)(53 137)(54 138)(55 139)(56 140)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 113)(72 114)(73 95)(74 96)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 127)(82 128)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(97 131)(98 132)(99 133)(100 134)(101 135)(102 136)(103 129)(104 130)
G:=sub<Sym(144)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,97)(17,87)(18,88)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(33,77)(34,78)(35,79)(36,80)(37,73)(38,74)(39,75)(40,76)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,129)(49,119)(50,120)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,127)(58,128)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(89,108)(90,109)(91,110)(92,111)(93,112)(94,105)(95,106)(96,107), (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), (1,104,130)(2,97,131)(3,98,132)(4,99,133)(5,100,134)(6,101,135)(7,102,136)(8,103,129)(9,43,29)(10,44,30)(11,45,31)(12,46,32)(13,47,25)(14,48,26)(15,41,27)(16,42,28)(17,56,39)(18,49,40)(19,50,33)(20,51,34)(21,52,35)(22,53,36)(23,54,37)(24,55,38)(57,110,142)(58,111,143)(59,112,144)(60,105,137)(61,106,138)(62,107,139)(63,108,140)(64,109,141)(65,122,94)(66,123,95)(67,124,96)(68,125,89)(69,126,90)(70,127,91)(71,128,92)(72,121,93)(73,85,116)(74,86,117)(75,87,118)(76,88,119)(77,81,120)(78,82,113)(79,83,114)(80,84,115), (1,118,68)(2,119,69)(3,120,70)(4,113,71)(5,114,72)(6,115,65)(7,116,66)(8,117,67)(9,33,57)(10,34,58)(11,35,59)(12,36,60)(13,37,61)(14,38,62)(15,39,63)(16,40,64)(17,108,41)(18,109,42)(19,110,43)(20,111,44)(21,112,45)(22,105,46)(23,106,47)(24,107,48)(25,54,138)(26,55,139)(27,56,140)(28,49,141)(29,50,142)(30,51,143)(31,52,144)(32,53,137)(73,123,102)(74,124,103)(75,125,104)(76,126,97)(77,127,98)(78,128,99)(79,121,100)(80,122,101)(81,91,132)(82,92,133)(83,93,134)(84,94,135)(85,95,136)(86,96,129)(87,89,130)(88,90,131), (9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(33,110)(34,111)(35,112)(36,105)(37,106)(38,107)(39,108)(40,109)(49,141)(50,142)(51,143)(52,144)(53,137)(54,138)(55,139)(56,140)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,113)(72,114)(73,95)(74,96)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,127)(82,128)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)(103,129)(104,130)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,97)(17,87)(18,88)(19,81)(20,82)(21,83)(22,84)(23,85)(24,86)(33,77)(34,78)(35,79)(36,80)(37,73)(38,74)(39,75)(40,76)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,129)(49,119)(50,120)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,127)(58,128)(59,121)(60,122)(61,123)(62,124)(63,125)(64,126)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(89,108)(90,109)(91,110)(92,111)(93,112)(94,105)(95,106)(96,107), (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), (1,104,130)(2,97,131)(3,98,132)(4,99,133)(5,100,134)(6,101,135)(7,102,136)(8,103,129)(9,43,29)(10,44,30)(11,45,31)(12,46,32)(13,47,25)(14,48,26)(15,41,27)(16,42,28)(17,56,39)(18,49,40)(19,50,33)(20,51,34)(21,52,35)(22,53,36)(23,54,37)(24,55,38)(57,110,142)(58,111,143)(59,112,144)(60,105,137)(61,106,138)(62,107,139)(63,108,140)(64,109,141)(65,122,94)(66,123,95)(67,124,96)(68,125,89)(69,126,90)(70,127,91)(71,128,92)(72,121,93)(73,85,116)(74,86,117)(75,87,118)(76,88,119)(77,81,120)(78,82,113)(79,83,114)(80,84,115), (1,118,68)(2,119,69)(3,120,70)(4,113,71)(5,114,72)(6,115,65)(7,116,66)(8,117,67)(9,33,57)(10,34,58)(11,35,59)(12,36,60)(13,37,61)(14,38,62)(15,39,63)(16,40,64)(17,108,41)(18,109,42)(19,110,43)(20,111,44)(21,112,45)(22,105,46)(23,106,47)(24,107,48)(25,54,138)(26,55,139)(27,56,140)(28,49,141)(29,50,142)(30,51,143)(31,52,144)(32,53,137)(73,123,102)(74,124,103)(75,125,104)(76,126,97)(77,127,98)(78,128,99)(79,121,100)(80,122,101)(81,91,132)(82,92,133)(83,93,134)(84,94,135)(85,95,136)(86,96,129)(87,89,130)(88,90,131), (9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(33,110)(34,111)(35,112)(36,105)(37,106)(38,107)(39,108)(40,109)(49,141)(50,142)(51,143)(52,144)(53,137)(54,138)(55,139)(56,140)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,113)(72,114)(73,95)(74,96)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,127)(82,128)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136)(103,129)(104,130) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,97),(17,87),(18,88),(19,81),(20,82),(21,83),(22,84),(23,85),(24,86),(33,77),(34,78),(35,79),(36,80),(37,73),(38,74),(39,75),(40,76),(41,130),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,129),(49,119),(50,120),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,127),(58,128),(59,121),(60,122),(61,123),(62,124),(63,125),(64,126),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144),(89,108),(90,109),(91,110),(92,111),(93,112),(94,105),(95,106),(96,107)], [(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)], [(1,104,130),(2,97,131),(3,98,132),(4,99,133),(5,100,134),(6,101,135),(7,102,136),(8,103,129),(9,43,29),(10,44,30),(11,45,31),(12,46,32),(13,47,25),(14,48,26),(15,41,27),(16,42,28),(17,56,39),(18,49,40),(19,50,33),(20,51,34),(21,52,35),(22,53,36),(23,54,37),(24,55,38),(57,110,142),(58,111,143),(59,112,144),(60,105,137),(61,106,138),(62,107,139),(63,108,140),(64,109,141),(65,122,94),(66,123,95),(67,124,96),(68,125,89),(69,126,90),(70,127,91),(71,128,92),(72,121,93),(73,85,116),(74,86,117),(75,87,118),(76,88,119),(77,81,120),(78,82,113),(79,83,114),(80,84,115)], [(1,118,68),(2,119,69),(3,120,70),(4,113,71),(5,114,72),(6,115,65),(7,116,66),(8,117,67),(9,33,57),(10,34,58),(11,35,59),(12,36,60),(13,37,61),(14,38,62),(15,39,63),(16,40,64),(17,108,41),(18,109,42),(19,110,43),(20,111,44),(21,112,45),(22,105,46),(23,106,47),(24,107,48),(25,54,138),(26,55,139),(27,56,140),(28,49,141),(29,50,142),(30,51,143),(31,52,144),(32,53,137),(73,123,102),(74,124,103),(75,125,104),(76,126,97),(77,127,98),(78,128,99),(79,121,100),(80,122,101),(81,91,132),(82,92,133),(83,93,134),(84,94,135),(85,95,136),(86,96,129),(87,89,130),(88,90,131)], [(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,41),(16,42),(17,63),(18,64),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(33,110),(34,111),(35,112),(36,105),(37,106),(38,107),(39,108),(40,109),(49,141),(50,142),(51,143),(52,144),(53,137),(54,138),(55,139),(56,140),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,113),(72,114),(73,95),(74,96),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,127),(82,128),(83,121),(84,122),(85,123),(86,124),(87,125),(88,126),(97,131),(98,132),(99,133),(100,134),(101,135),(102,136),(103,129),(104,130)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6L | 8A | ··· | 8H | 8I | ··· | 8P | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | D6 | D6 | C4×S3 | C4×S3 | S3×C8 |
| kernel | C2×C8×C3⋊S3 | C8×C3⋊S3 | C2×C32⋊4C8 | C6×C24 | C2×C4×C3⋊S3 | C4×C3⋊S3 | C2×C3⋊Dic3 | C22×C3⋊S3 | C2×C3⋊S3 | C2×C24 | C24 | C2×C12 | C12 | C2×C6 | C6 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 16 | 4 | 8 | 4 | 8 | 8 | 32 |
Matrix representation of C2×C8×C3⋊S3 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 63 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[10,0,0,0,0,10,0,0,0,0,63,0,0,0,0,63],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;
C2×C8×C3⋊S3 in GAP, Magma, Sage, TeX
C_2\times C_8\times C_3\rtimes S_3
% in TeX
G:=Group("C2xC8xC3:S3"); // GroupNames label
G:=SmallGroup(288,756);
// by ID
G=gap.SmallGroup(288,756);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,58,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations