metabelian, supersoluble, monomial
Aliases: C48⋊6S3, C24.92D6, C32⋊8M5(2), (C3×C48)⋊9C2, C16⋊3(C3⋊S3), C6.13(S3×C8), C12.76(C4×S3), C3⋊2(D6.C8), C3⋊Dic3.5C8, C24.S3⋊8C2, C32⋊4C8.6C4, (C3×C24).69C22, C2.3(C8×C3⋊S3), (C4×C3⋊S3).9C4, (C8×C3⋊S3).5C2, (C2×C3⋊S3).5C8, C8.19(C2×C3⋊S3), C4.17(C4×C3⋊S3), (C3×C6).33(C2×C8), (C3×C12).108(C2×C4), SmallGroup(288,273)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C48⋊S3
G = < a,b,c | a48=b3=c2=1, ab=ba, cac=a41, cbc=b-1 >
Subgroups: 228 in 78 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, C32, Dic3, C12, D6, C16, C16, C2×C8, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, M5(2), C3⋊Dic3, C3×C12, C2×C3⋊S3, C3⋊C16, C48, S3×C8, C32⋊4C8, C3×C24, C4×C3⋊S3, D6.C8, C24.S3, C3×C48, C8×C3⋊S3, C48⋊S3
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, C3⋊S3, C4×S3, M5(2), C2×C3⋊S3, S3×C8, C4×C3⋊S3, D6.C8, C8×C3⋊S3, C48⋊S3
►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 87 138)(2 88 139)(3 89 140)(4 90 141)(5 91 142)(6 92 143)(7 93 144)(8 94 97)(9 95 98)(10 96 99)(11 49 100)(12 50 101)(13 51 102)(14 52 103)(15 53 104)(16 54 105)(17 55 106)(18 56 107)(19 57 108)(20 58 109)(21 59 110)(22 60 111)(23 61 112)(24 62 113)(25 63 114)(26 64 115)(27 65 116)(28 66 117)(29 67 118)(30 68 119)(31 69 120)(32 70 121)(33 71 122)(34 72 123)(35 73 124)(36 74 125)(37 75 126)(38 76 127)(39 77 128)(40 78 129)(41 79 130)(42 80 131)(43 81 132)(44 82 133)(45 83 134)(46 84 135)(47 85 136)(48 86 137)
(2 42)(3 35)(4 28)(5 21)(6 14)(8 48)(9 41)(10 34)(11 27)(12 20)(15 47)(16 40)(17 33)(18 26)(22 46)(23 39)(24 32)(29 45)(30 38)(36 44)(49 116)(50 109)(51 102)(52 143)(53 136)(54 129)(55 122)(56 115)(57 108)(58 101)(59 142)(60 135)(61 128)(62 121)(63 114)(64 107)(65 100)(66 141)(67 134)(68 127)(69 120)(70 113)(71 106)(72 99)(73 140)(74 133)(75 126)(76 119)(77 112)(78 105)(79 98)(80 139)(81 132)(82 125)(83 118)(84 111)(85 104)(86 97)(87 138)(88 131)(89 124)(90 117)(91 110)(92 103)(93 144)(94 137)(95 130)(96 123)
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), (1,87,138)(2,88,139)(3,89,140)(4,90,141)(5,91,142)(6,92,143)(7,93,144)(8,94,97)(9,95,98)(10,96,99)(11,49,100)(12,50,101)(13,51,102)(14,52,103)(15,53,104)(16,54,105)(17,55,106)(18,56,107)(19,57,108)(20,58,109)(21,59,110)(22,60,111)(23,61,112)(24,62,113)(25,63,114)(26,64,115)(27,65,116)(28,66,117)(29,67,118)(30,68,119)(31,69,120)(32,70,121)(33,71,122)(34,72,123)(35,73,124)(36,74,125)(37,75,126)(38,76,127)(39,77,128)(40,78,129)(41,79,130)(42,80,131)(43,81,132)(44,82,133)(45,83,134)(46,84,135)(47,85,136)(48,86,137), (2,42)(3,35)(4,28)(5,21)(6,14)(8,48)(9,41)(10,34)(11,27)(12,20)(15,47)(16,40)(17,33)(18,26)(22,46)(23,39)(24,32)(29,45)(30,38)(36,44)(49,116)(50,109)(51,102)(52,143)(53,136)(54,129)(55,122)(56,115)(57,108)(58,101)(59,142)(60,135)(61,128)(62,121)(63,114)(64,107)(65,100)(66,141)(67,134)(68,127)(69,120)(70,113)(71,106)(72,99)(73,140)(74,133)(75,126)(76,119)(77,112)(78,105)(79,98)(80,139)(81,132)(82,125)(83,118)(84,111)(85,104)(86,97)(87,138)(88,131)(89,124)(90,117)(91,110)(92,103)(93,144)(94,137)(95,130)(96,123)>;
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), (1,87,138)(2,88,139)(3,89,140)(4,90,141)(5,91,142)(6,92,143)(7,93,144)(8,94,97)(9,95,98)(10,96,99)(11,49,100)(12,50,101)(13,51,102)(14,52,103)(15,53,104)(16,54,105)(17,55,106)(18,56,107)(19,57,108)(20,58,109)(21,59,110)(22,60,111)(23,61,112)(24,62,113)(25,63,114)(26,64,115)(27,65,116)(28,66,117)(29,67,118)(30,68,119)(31,69,120)(32,70,121)(33,71,122)(34,72,123)(35,73,124)(36,74,125)(37,75,126)(38,76,127)(39,77,128)(40,78,129)(41,79,130)(42,80,131)(43,81,132)(44,82,133)(45,83,134)(46,84,135)(47,85,136)(48,86,137), (2,42)(3,35)(4,28)(5,21)(6,14)(8,48)(9,41)(10,34)(11,27)(12,20)(15,47)(16,40)(17,33)(18,26)(22,46)(23,39)(24,32)(29,45)(30,38)(36,44)(49,116)(50,109)(51,102)(52,143)(53,136)(54,129)(55,122)(56,115)(57,108)(58,101)(59,142)(60,135)(61,128)(62,121)(63,114)(64,107)(65,100)(66,141)(67,134)(68,127)(69,120)(70,113)(71,106)(72,99)(73,140)(74,133)(75,126)(76,119)(77,112)(78,105)(79,98)(80,139)(81,132)(82,125)(83,118)(84,111)(85,104)(86,97)(87,138)(88,131)(89,124)(90,117)(91,110)(92,103)(93,144)(94,137)(95,130)(96,123) );
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)], [(1,87,138),(2,88,139),(3,89,140),(4,90,141),(5,91,142),(6,92,143),(7,93,144),(8,94,97),(9,95,98),(10,96,99),(11,49,100),(12,50,101),(13,51,102),(14,52,103),(15,53,104),(16,54,105),(17,55,106),(18,56,107),(19,57,108),(20,58,109),(21,59,110),(22,60,111),(23,61,112),(24,62,113),(25,63,114),(26,64,115),(27,65,116),(28,66,117),(29,67,118),(30,68,119),(31,69,120),(32,70,121),(33,71,122),(34,72,123),(35,73,124),(36,74,125),(37,75,126),(38,76,127),(39,77,128),(40,78,129),(41,79,130),(42,80,131),(43,81,132),(44,82,133),(45,83,134),(46,84,135),(47,85,136),(48,86,137)], [(2,42),(3,35),(4,28),(5,21),(6,14),(8,48),(9,41),(10,34),(11,27),(12,20),(15,47),(16,40),(17,33),(18,26),(22,46),(23,39),(24,32),(29,45),(30,38),(36,44),(49,116),(50,109),(51,102),(52,143),(53,136),(54,129),(55,122),(56,115),(57,108),(58,101),(59,142),(60,135),(61,128),(62,121),(63,114),(64,107),(65,100),(66,141),(67,134),(68,127),(69,120),(70,113),(71,106),(72,99),(73,140),(74,133),(75,126),(76,119),(77,112),(78,105),(79,98),(80,139),(81,132),(82,125),(83,118),(84,111),(85,104),(86,97),(87,138),(88,131),(89,124),(90,117),(91,110),(92,103),(93,144),(94,137),(95,130),(96,123)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 12A | ··· | 12H | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | ··· | 24P | 48A | ··· | 48AF |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 18 | 18 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D6 | C4×S3 | M5(2) | S3×C8 | D6.C8 |
| kernel | C48⋊S3 | C24.S3 | C3×C48 | C8×C3⋊S3 | C32⋊4C8 | C4×C3⋊S3 | C3⋊Dic3 | C2×C3⋊S3 | C48 | C24 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 16 | 32 |
Matrix representation of C48⋊S3 ►in GL6(𝔽97)
| 47 | 47 | 0 | 0 | 0 | 0 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 3 | 0 | 0 |
| 0 | 0 | 83 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 75 | 0 |
| 0 | 0 | 0 | 0 | 0 | 75 |
| 96 | 96 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 96 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 96 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(97))| [47,50,0,0,0,0,47,0,0,0,0,0,0,0,50,83,0,0,0,0,3,47,0,0,0,0,0,0,75,0,0,0,0,0,0,75],[96,1,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,1,0,0,0,0,96,0],[1,96,0,0,0,0,0,96,0,0,0,0,0,0,1,96,0,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C48⋊S3 in GAP, Magma, Sage, TeX
C_{48}\rtimes S_3 % in TeX
G:=Group("C48:S3"); // GroupNames label
G:=SmallGroup(288,273);
// by ID
G=gap.SmallGroup(288,273);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,36,58,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^48=b^3=c^2=1,a*b=b*a,c*a*c=a^41,c*b*c=b^-1>;
// generators/relations