direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C24.S3, C33⋊6C16, C32⋊6C48, C24.23(C3×S3), C32⋊5(C3⋊C16), (C3×C24).22C6, (C3×C6).10C24, (C3×C24).25S3, (C32×C6).6C8, C24.16(C3⋊S3), (C3×C12).23C12, (C32×C24).6C2, (C32×C12).11C4, C6.6(C32⋊4C8), C12.17(C3×Dic3), (C3×C12).27Dic3, C12.15(C3⋊Dic3), C3⋊(C3×C3⋊C16), C6.7(C3×C3⋊C8), C8.2(C3×C3⋊S3), C2.(C3×C32⋊4C8), (C3×C6).11(C3⋊C8), C4.2(C3×C3⋊Dic3), SmallGroup(432,230)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3×C24.S3 |
Generators and relations for C3×C24.S3
G = < a,b,c,d | a3=b24=c3=1, d2=b9, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b17, dcd-1=c-1 >
Subgroups: 156 in 92 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C8, C32, C32, C32, C12, C12, C12, C16, C3×C6, C3×C6, C3×C6, C24, C24, C24, C33, C3×C12, C3×C12, C3×C12, C3⋊C16, C48, C32×C6, C3×C24, C3×C24, C3×C24, C32×C12, C3×C3⋊C16, C24.S3, C32×C24, C3×C24.S3
Quotients: C1, C2, C3, C4, S3, C6, C8, Dic3, C12, C16, C3×S3, C3⋊S3, C3⋊C8, C24, C3×Dic3, C3⋊Dic3, C3⋊C16, C48, C3×C3⋊S3, C3×C3⋊C8, C32⋊4C8, C3×C3⋊Dic3, C3×C3⋊C16, C24.S3, C3×C32⋊4C8, C3×C24.S3
►On 144 points
Generators in S144
(1 131 82)(2 132 83)(3 133 84)(4 134 85)(5 135 86)(6 136 87)(7 137 88)(8 138 89)(9 139 90)(10 140 91)(11 141 92)(12 142 93)(13 143 94)(14 144 95)(15 121 96)(16 122 73)(17 123 74)(18 124 75)(19 125 76)(20 126 77)(21 127 78)(22 128 79)(23 129 80)(24 130 81)(25 64 109)(26 65 110)(27 66 111)(28 67 112)(29 68 113)(30 69 114)(31 70 115)(32 71 116)(33 72 117)(34 49 118)(35 50 119)(36 51 120)(37 52 97)(38 53 98)(39 54 99)(40 55 100)(41 56 101)(42 57 102)(43 58 103)(44 59 104)(45 60 105)(46 61 106)(47 62 107)(48 63 108)
(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)
(1 131 82)(2 132 83)(3 133 84)(4 134 85)(5 135 86)(6 136 87)(7 137 88)(8 138 89)(9 139 90)(10 140 91)(11 141 92)(12 142 93)(13 143 94)(14 144 95)(15 121 96)(16 122 73)(17 123 74)(18 124 75)(19 125 76)(20 126 77)(21 127 78)(22 128 79)(23 129 80)(24 130 81)(25 109 64)(26 110 65)(27 111 66)(28 112 67)(29 113 68)(30 114 69)(31 115 70)(32 116 71)(33 117 72)(34 118 49)(35 119 50)(36 120 51)(37 97 52)(38 98 53)(39 99 54)(40 100 55)(41 101 56)(42 102 57)(43 103 58)(44 104 59)(45 105 60)(46 106 61)(47 107 62)(48 108 63)
(1 38 10 47 19 32 4 41 13 26 22 35 7 44 16 29)(2 31 11 40 20 25 5 34 14 43 23 28 8 37 17 46)(3 48 12 33 21 42 6 27 15 36 24 45 9 30 18 39)(49 144 58 129 67 138 52 123 61 132 70 141 55 126 64 135)(50 137 59 122 68 131 53 140 62 125 71 134 56 143 65 128)(51 130 60 139 69 124 54 133 63 142 72 127 57 136 66 121)(73 113 82 98 91 107 76 116 85 101 94 110 79 119 88 104)(74 106 83 115 92 100 77 109 86 118 95 103 80 112 89 97)(75 99 84 108 93 117 78 102 87 111 96 120 81 105 90 114)
G:=sub<Sym(144)| (1,131,82)(2,132,83)(3,133,84)(4,134,85)(5,135,86)(6,136,87)(7,137,88)(8,138,89)(9,139,90)(10,140,91)(11,141,92)(12,142,93)(13,143,94)(14,144,95)(15,121,96)(16,122,73)(17,123,74)(18,124,75)(19,125,76)(20,126,77)(21,127,78)(22,128,79)(23,129,80)(24,130,81)(25,64,109)(26,65,110)(27,66,111)(28,67,112)(29,68,113)(30,69,114)(31,70,115)(32,71,116)(33,72,117)(34,49,118)(35,50,119)(36,51,120)(37,52,97)(38,53,98)(39,54,99)(40,55,100)(41,56,101)(42,57,102)(43,58,103)(44,59,104)(45,60,105)(46,61,106)(47,62,107)(48,63,108), (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), (1,131,82)(2,132,83)(3,133,84)(4,134,85)(5,135,86)(6,136,87)(7,137,88)(8,138,89)(9,139,90)(10,140,91)(11,141,92)(12,142,93)(13,143,94)(14,144,95)(15,121,96)(16,122,73)(17,123,74)(18,124,75)(19,125,76)(20,126,77)(21,127,78)(22,128,79)(23,129,80)(24,130,81)(25,109,64)(26,110,65)(27,111,66)(28,112,67)(29,113,68)(30,114,69)(31,115,70)(32,116,71)(33,117,72)(34,118,49)(35,119,50)(36,120,51)(37,97,52)(38,98,53)(39,99,54)(40,100,55)(41,101,56)(42,102,57)(43,103,58)(44,104,59)(45,105,60)(46,106,61)(47,107,62)(48,108,63), (1,38,10,47,19,32,4,41,13,26,22,35,7,44,16,29)(2,31,11,40,20,25,5,34,14,43,23,28,8,37,17,46)(3,48,12,33,21,42,6,27,15,36,24,45,9,30,18,39)(49,144,58,129,67,138,52,123,61,132,70,141,55,126,64,135)(50,137,59,122,68,131,53,140,62,125,71,134,56,143,65,128)(51,130,60,139,69,124,54,133,63,142,72,127,57,136,66,121)(73,113,82,98,91,107,76,116,85,101,94,110,79,119,88,104)(74,106,83,115,92,100,77,109,86,118,95,103,80,112,89,97)(75,99,84,108,93,117,78,102,87,111,96,120,81,105,90,114)>;
G:=Group( (1,131,82)(2,132,83)(3,133,84)(4,134,85)(5,135,86)(6,136,87)(7,137,88)(8,138,89)(9,139,90)(10,140,91)(11,141,92)(12,142,93)(13,143,94)(14,144,95)(15,121,96)(16,122,73)(17,123,74)(18,124,75)(19,125,76)(20,126,77)(21,127,78)(22,128,79)(23,129,80)(24,130,81)(25,64,109)(26,65,110)(27,66,111)(28,67,112)(29,68,113)(30,69,114)(31,70,115)(32,71,116)(33,72,117)(34,49,118)(35,50,119)(36,51,120)(37,52,97)(38,53,98)(39,54,99)(40,55,100)(41,56,101)(42,57,102)(43,58,103)(44,59,104)(45,60,105)(46,61,106)(47,62,107)(48,63,108), (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), (1,131,82)(2,132,83)(3,133,84)(4,134,85)(5,135,86)(6,136,87)(7,137,88)(8,138,89)(9,139,90)(10,140,91)(11,141,92)(12,142,93)(13,143,94)(14,144,95)(15,121,96)(16,122,73)(17,123,74)(18,124,75)(19,125,76)(20,126,77)(21,127,78)(22,128,79)(23,129,80)(24,130,81)(25,109,64)(26,110,65)(27,111,66)(28,112,67)(29,113,68)(30,114,69)(31,115,70)(32,116,71)(33,117,72)(34,118,49)(35,119,50)(36,120,51)(37,97,52)(38,98,53)(39,99,54)(40,100,55)(41,101,56)(42,102,57)(43,103,58)(44,104,59)(45,105,60)(46,106,61)(47,107,62)(48,108,63), (1,38,10,47,19,32,4,41,13,26,22,35,7,44,16,29)(2,31,11,40,20,25,5,34,14,43,23,28,8,37,17,46)(3,48,12,33,21,42,6,27,15,36,24,45,9,30,18,39)(49,144,58,129,67,138,52,123,61,132,70,141,55,126,64,135)(50,137,59,122,68,131,53,140,62,125,71,134,56,143,65,128)(51,130,60,139,69,124,54,133,63,142,72,127,57,136,66,121)(73,113,82,98,91,107,76,116,85,101,94,110,79,119,88,104)(74,106,83,115,92,100,77,109,86,118,95,103,80,112,89,97)(75,99,84,108,93,117,78,102,87,111,96,120,81,105,90,114) );
G=PermutationGroup([[(1,131,82),(2,132,83),(3,133,84),(4,134,85),(5,135,86),(6,136,87),(7,137,88),(8,138,89),(9,139,90),(10,140,91),(11,141,92),(12,142,93),(13,143,94),(14,144,95),(15,121,96),(16,122,73),(17,123,74),(18,124,75),(19,125,76),(20,126,77),(21,127,78),(22,128,79),(23,129,80),(24,130,81),(25,64,109),(26,65,110),(27,66,111),(28,67,112),(29,68,113),(30,69,114),(31,70,115),(32,71,116),(33,72,117),(34,49,118),(35,50,119),(36,51,120),(37,52,97),(38,53,98),(39,54,99),(40,55,100),(41,56,101),(42,57,102),(43,58,103),(44,59,104),(45,60,105),(46,61,106),(47,62,107),(48,63,108)], [(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)], [(1,131,82),(2,132,83),(3,133,84),(4,134,85),(5,135,86),(6,136,87),(7,137,88),(8,138,89),(9,139,90),(10,140,91),(11,141,92),(12,142,93),(13,143,94),(14,144,95),(15,121,96),(16,122,73),(17,123,74),(18,124,75),(19,125,76),(20,126,77),(21,127,78),(22,128,79),(23,129,80),(24,130,81),(25,109,64),(26,110,65),(27,111,66),(28,112,67),(29,113,68),(30,114,69),(31,115,70),(32,116,71),(33,117,72),(34,118,49),(35,119,50),(36,120,51),(37,97,52),(38,98,53),(39,99,54),(40,100,55),(41,101,56),(42,102,57),(43,103,58),(44,104,59),(45,105,60),(46,106,61),(47,107,62),(48,108,63)], [(1,38,10,47,19,32,4,41,13,26,22,35,7,44,16,29),(2,31,11,40,20,25,5,34,14,43,23,28,8,37,17,46),(3,48,12,33,21,42,6,27,15,36,24,45,9,30,18,39),(49,144,58,129,67,138,52,123,61,132,70,141,55,126,64,135),(50,137,59,122,68,131,53,140,62,125,71,134,56,143,65,128),(51,130,60,139,69,124,54,133,63,142,72,127,57,136,66,121),(73,113,82,98,91,107,76,116,85,101,94,110,79,119,88,104),(74,106,83,115,92,100,77,109,86,118,95,103,80,112,89,97),(75,99,84,108,93,117,78,102,87,111,96,120,81,105,90,114)]])
144 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 6A | 6B | 6C | ··· | 6N | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12AB | 16A | ··· | 16H | 24A | ··· | 24H | 24I | ··· | 24BD | 48A | ··· | 48P |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C16 | C24 | C48 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3⋊C16 | C3×C3⋊C8 | C3×C3⋊C16 |
| kernel | C3×C24.S3 | C32×C24 | C24.S3 | C32×C12 | C3×C24 | C32×C6 | C3×C12 | C33 | C3×C6 | C32 | C3×C24 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C3×C24.S3 ►in GL4(𝔽97) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 0 | 61 |
| 66 | 57 | 0 | 0 |
| 55 | 64 | 0 | 0 |
| 0 | 0 | 73 | 0 |
| 0 | 0 | 30 | 88 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 61 | 0 |
| 0 | 0 | 45 | 35 |
| 17 | 96 | 0 | 0 |
| 31 | 80 | 0 | 0 |
| 0 | 0 | 68 | 34 |
| 0 | 0 | 39 | 29 |
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,61,0,0,0,0,61],[66,55,0,0,57,64,0,0,0,0,73,30,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,61,45,0,0,0,35],[17,31,0,0,96,80,0,0,0,0,68,39,0,0,34,29] >;
C3×C24.S3 in GAP, Magma, Sage, TeX
C_3\times C_{24}.S_3 % in TeX
G:=Group("C3xC24.S3"); // GroupNames label
G:=SmallGroup(432,230);
// by ID
G=gap.SmallGroup(432,230);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,-2,-3,-3,42,58,80,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^24=c^3=1,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^17,d*c*d^-1=c^-1>;
// generators/relations