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 [×4], C4, C4, C22, S3 [×4], C6 [×4], C8, C8, C2×C4, C32, Dic3 [×4], C12 [×4], D6 [×4], C16, C16, C2×C8, C3⋊S3, C3×C6, C3⋊C8 [×4], C24 [×4], C4×S3 [×4], M5(2), C3⋊Dic3, C3×C12, C2×C3⋊S3, C3⋊C16 [×4], C48 [×4], S3×C8 [×4], C32⋊4C8, C3×C24, C4×C3⋊S3, D6.C8 [×4], C24.S3, C3×C48, C8×C3⋊S3, C48⋊S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, D6 [×4], C2×C8, C3⋊S3, C4×S3 [×4], M5(2), C2×C3⋊S3, S3×C8 [×4], C4×C3⋊S3, D6.C8 [×4], 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)
(1 120 87)(2 121 88)(3 122 89)(4 123 90)(5 124 91)(6 125 92)(7 126 93)(8 127 94)(9 128 95)(10 129 96)(11 130 49)(12 131 50)(13 132 51)(14 133 52)(15 134 53)(16 135 54)(17 136 55)(18 137 56)(19 138 57)(20 139 58)(21 140 59)(22 141 60)(23 142 61)(24 143 62)(25 144 63)(26 97 64)(27 98 65)(28 99 66)(29 100 67)(30 101 68)(31 102 69)(32 103 70)(33 104 71)(34 105 72)(35 106 73)(36 107 74)(37 108 75)(38 109 76)(39 110 77)(40 111 78)(41 112 79)(42 113 80)(43 114 81)(44 115 82)(45 116 83)(46 117 84)(47 118 85)(48 119 86)
(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 98)(50 139)(51 132)(52 125)(53 118)(54 111)(55 104)(56 97)(57 138)(58 131)(59 124)(60 117)(61 110)(62 103)(63 144)(64 137)(65 130)(66 123)(67 116)(68 109)(69 102)(70 143)(71 136)(72 129)(73 122)(74 115)(75 108)(76 101)(77 142)(78 135)(79 128)(80 121)(81 114)(82 107)(83 100)(84 141)(85 134)(86 127)(87 120)(88 113)(89 106)(90 99)(91 140)(92 133)(93 126)(94 119)(95 112)(96 105)
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), (1,120,87)(2,121,88)(3,122,89)(4,123,90)(5,124,91)(6,125,92)(7,126,93)(8,127,94)(9,128,95)(10,129,96)(11,130,49)(12,131,50)(13,132,51)(14,133,52)(15,134,53)(16,135,54)(17,136,55)(18,137,56)(19,138,57)(20,139,58)(21,140,59)(22,141,60)(23,142,61)(24,143,62)(25,144,63)(26,97,64)(27,98,65)(28,99,66)(29,100,67)(30,101,68)(31,102,69)(32,103,70)(33,104,71)(34,105,72)(35,106,73)(36,107,74)(37,108,75)(38,109,76)(39,110,77)(40,111,78)(41,112,79)(42,113,80)(43,114,81)(44,115,82)(45,116,83)(46,117,84)(47,118,85)(48,119,86), (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)(49,98)(50,139)(51,132)(52,125)(53,118)(54,111)(55,104)(56,97)(57,138)(58,131)(59,124)(60,117)(61,110)(62,103)(63,144)(64,137)(65,130)(66,123)(67,116)(68,109)(69,102)(70,143)(71,136)(72,129)(73,122)(74,115)(75,108)(76,101)(77,142)(78,135)(79,128)(80,121)(81,114)(82,107)(83,100)(84,141)(85,134)(86,127)(87,120)(88,113)(89,106)(90,99)(91,140)(92,133)(93,126)(94,119)(95,112)(96,105)>;
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), (1,120,87)(2,121,88)(3,122,89)(4,123,90)(5,124,91)(6,125,92)(7,126,93)(8,127,94)(9,128,95)(10,129,96)(11,130,49)(12,131,50)(13,132,51)(14,133,52)(15,134,53)(16,135,54)(17,136,55)(18,137,56)(19,138,57)(20,139,58)(21,140,59)(22,141,60)(23,142,61)(24,143,62)(25,144,63)(26,97,64)(27,98,65)(28,99,66)(29,100,67)(30,101,68)(31,102,69)(32,103,70)(33,104,71)(34,105,72)(35,106,73)(36,107,74)(37,108,75)(38,109,76)(39,110,77)(40,111,78)(41,112,79)(42,113,80)(43,114,81)(44,115,82)(45,116,83)(46,117,84)(47,118,85)(48,119,86), (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)(49,98)(50,139)(51,132)(52,125)(53,118)(54,111)(55,104)(56,97)(57,138)(58,131)(59,124)(60,117)(61,110)(62,103)(63,144)(64,137)(65,130)(66,123)(67,116)(68,109)(69,102)(70,143)(71,136)(72,129)(73,122)(74,115)(75,108)(76,101)(77,142)(78,135)(79,128)(80,121)(81,114)(82,107)(83,100)(84,141)(85,134)(86,127)(87,120)(88,113)(89,106)(90,99)(91,140)(92,133)(93,126)(94,119)(95,112)(96,105) );
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)], [(1,120,87),(2,121,88),(3,122,89),(4,123,90),(5,124,91),(6,125,92),(7,126,93),(8,127,94),(9,128,95),(10,129,96),(11,130,49),(12,131,50),(13,132,51),(14,133,52),(15,134,53),(16,135,54),(17,136,55),(18,137,56),(19,138,57),(20,139,58),(21,140,59),(22,141,60),(23,142,61),(24,143,62),(25,144,63),(26,97,64),(27,98,65),(28,99,66),(29,100,67),(30,101,68),(31,102,69),(32,103,70),(33,104,71),(34,105,72),(35,106,73),(36,107,74),(37,108,75),(38,109,76),(39,110,77),(40,111,78),(41,112,79),(42,113,80),(43,114,81),(44,115,82),(45,116,83),(46,117,84),(47,118,85),(48,119,86)], [(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),(49,98),(50,139),(51,132),(52,125),(53,118),(54,111),(55,104),(56,97),(57,138),(58,131),(59,124),(60,117),(61,110),(62,103),(63,144),(64,137),(65,130),(66,123),(67,116),(68,109),(69,102),(70,143),(71,136),(72,129),(73,122),(74,115),(75,108),(76,101),(77,142),(78,135),(79,128),(80,121),(81,114),(82,107),(83,100),(84,141),(85,134),(86,127),(87,120),(88,113),(89,106),(90,99),(91,140),(92,133),(93,126),(94,119),(95,112),(96,105)])
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