metabelian, supersoluble, monomial, A-group
Aliases: C24.9S3, C32⋊4C16, C12.7Dic3, C3⋊(C3⋊C16), C6.3(C3⋊C8), (C3×C6).4C8, C8.2(C3⋊S3), (C3×C24).5C2, (C3×C12).7C4, C2.(C32⋊4C8), C4.2(C3⋊Dic3), SmallGroup(144,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C24.S3 |
Generators and relations for C24.S3
G = < a,b,c | a24=b3=1, c2=a9, ab=ba, cac-1=a17, cbc-1=b-1 >
9C16
Smallest permutation representation of C24.S3
►Regular action 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 58 142)(2 59 143)(3 60 144)(4 61 121)(5 62 122)(6 63 123)(7 64 124)(8 65 125)(9 66 126)(10 67 127)(11 68 128)(12 69 129)(13 70 130)(14 71 131)(15 72 132)(16 49 133)(17 50 134)(18 51 135)(19 52 136)(20 53 137)(21 54 138)(22 55 139)(23 56 140)(24 57 141)(25 79 106)(26 80 107)(27 81 108)(28 82 109)(29 83 110)(30 84 111)(31 85 112)(32 86 113)(33 87 114)(34 88 115)(35 89 116)(36 90 117)(37 91 118)(38 92 119)(39 93 120)(40 94 97)(41 95 98)(42 96 99)(43 73 100)(44 74 101)(45 75 102)(46 76 103)(47 77 104)(48 78 105)
(1 79 10 88 19 73 4 82 13 91 22 76 7 85 16 94)(2 96 11 81 20 90 5 75 14 84 23 93 8 78 17 87)(3 89 12 74 21 83 6 92 15 77 24 86 9 95 18 80)(25 67 34 52 43 61 28 70 37 55 46 64 31 49 40 58)(26 60 35 69 44 54 29 63 38 72 47 57 32 66 41 51)(27 53 36 62 45 71 30 56 39 65 48 50 33 59 42 68)(97 142 106 127 115 136 100 121 109 130 118 139 103 124 112 133)(98 135 107 144 116 129 101 138 110 123 119 132 104 141 113 126)(99 128 108 137 117 122 102 131 111 140 120 125 105 134 114 143)
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), (1,58,142)(2,59,143)(3,60,144)(4,61,121)(5,62,122)(6,63,123)(7,64,124)(8,65,125)(9,66,126)(10,67,127)(11,68,128)(12,69,129)(13,70,130)(14,71,131)(15,72,132)(16,49,133)(17,50,134)(18,51,135)(19,52,136)(20,53,137)(21,54,138)(22,55,139)(23,56,140)(24,57,141)(25,79,106)(26,80,107)(27,81,108)(28,82,109)(29,83,110)(30,84,111)(31,85,112)(32,86,113)(33,87,114)(34,88,115)(35,89,116)(36,90,117)(37,91,118)(38,92,119)(39,93,120)(40,94,97)(41,95,98)(42,96,99)(43,73,100)(44,74,101)(45,75,102)(46,76,103)(47,77,104)(48,78,105), (1,79,10,88,19,73,4,82,13,91,22,76,7,85,16,94)(2,96,11,81,20,90,5,75,14,84,23,93,8,78,17,87)(3,89,12,74,21,83,6,92,15,77,24,86,9,95,18,80)(25,67,34,52,43,61,28,70,37,55,46,64,31,49,40,58)(26,60,35,69,44,54,29,63,38,72,47,57,32,66,41,51)(27,53,36,62,45,71,30,56,39,65,48,50,33,59,42,68)(97,142,106,127,115,136,100,121,109,130,118,139,103,124,112,133)(98,135,107,144,116,129,101,138,110,123,119,132,104,141,113,126)(99,128,108,137,117,122,102,131,111,140,120,125,105,134,114,143)>;
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), (1,58,142)(2,59,143)(3,60,144)(4,61,121)(5,62,122)(6,63,123)(7,64,124)(8,65,125)(9,66,126)(10,67,127)(11,68,128)(12,69,129)(13,70,130)(14,71,131)(15,72,132)(16,49,133)(17,50,134)(18,51,135)(19,52,136)(20,53,137)(21,54,138)(22,55,139)(23,56,140)(24,57,141)(25,79,106)(26,80,107)(27,81,108)(28,82,109)(29,83,110)(30,84,111)(31,85,112)(32,86,113)(33,87,114)(34,88,115)(35,89,116)(36,90,117)(37,91,118)(38,92,119)(39,93,120)(40,94,97)(41,95,98)(42,96,99)(43,73,100)(44,74,101)(45,75,102)(46,76,103)(47,77,104)(48,78,105), (1,79,10,88,19,73,4,82,13,91,22,76,7,85,16,94)(2,96,11,81,20,90,5,75,14,84,23,93,8,78,17,87)(3,89,12,74,21,83,6,92,15,77,24,86,9,95,18,80)(25,67,34,52,43,61,28,70,37,55,46,64,31,49,40,58)(26,60,35,69,44,54,29,63,38,72,47,57,32,66,41,51)(27,53,36,62,45,71,30,56,39,65,48,50,33,59,42,68)(97,142,106,127,115,136,100,121,109,130,118,139,103,124,112,133)(98,135,107,144,116,129,101,138,110,123,119,132,104,141,113,126)(99,128,108,137,117,122,102,131,111,140,120,125,105,134,114,143) );
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)], [(1,58,142),(2,59,143),(3,60,144),(4,61,121),(5,62,122),(6,63,123),(7,64,124),(8,65,125),(9,66,126),(10,67,127),(11,68,128),(12,69,129),(13,70,130),(14,71,131),(15,72,132),(16,49,133),(17,50,134),(18,51,135),(19,52,136),(20,53,137),(21,54,138),(22,55,139),(23,56,140),(24,57,141),(25,79,106),(26,80,107),(27,81,108),(28,82,109),(29,83,110),(30,84,111),(31,85,112),(32,86,113),(33,87,114),(34,88,115),(35,89,116),(36,90,117),(37,91,118),(38,92,119),(39,93,120),(40,94,97),(41,95,98),(42,96,99),(43,73,100),(44,74,101),(45,75,102),(46,76,103),(47,77,104),(48,78,105)], [(1,79,10,88,19,73,4,82,13,91,22,76,7,85,16,94),(2,96,11,81,20,90,5,75,14,84,23,93,8,78,17,87),(3,89,12,74,21,83,6,92,15,77,24,86,9,95,18,80),(25,67,34,52,43,61,28,70,37,55,46,64,31,49,40,58),(26,60,35,69,44,54,29,63,38,72,47,57,32,66,41,51),(27,53,36,62,45,71,30,56,39,65,48,50,33,59,42,68),(97,142,106,127,115,136,100,121,109,130,118,139,103,124,112,133),(98,135,107,144,116,129,101,138,110,123,119,132,104,141,113,126),(99,128,108,137,117,122,102,131,111,140,120,125,105,134,114,143)])
C24.S3 is a maximal subgroup of
C32⋊2C32 S3×C3⋊C16 C24.61D6 C32⋊2D16 D24.S3 C32⋊2Q32 C16×C3⋊S3 C48⋊S3 C24.94D6 C32⋊7D16 C32⋊8SD32 C32⋊10SD32 C32⋊7Q32 He3⋊3C16 C72.S3 C33⋊7C16
C24.S3 is a maximal quotient of
C48.S3 C72.S3 He3⋊4C16 C33⋊7C16
48 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 16A | ··· | 16H | 24A | ··· | 24P |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | S3 | Dic3 | C3⋊C8 | C3⋊C16 |
| kernel | C24.S3 | C3×C24 | C3×C12 | C3×C6 | C32 | C24 | C12 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 8 | 4 | 4 | 8 | 16 |
Matrix representation of C24.S3 ►in GL5(𝔽97)
| 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 |
| 0 | 75 | 75 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 96 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 96 | 96 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 96 |
| 0 | 0 | 0 | 1 | 0 |
| 85 | 0 | 0 | 0 | 0 |
| 0 | 28 | 10 | 0 | 0 |
| 0 | 79 | 69 | 0 | 0 |
| 0 | 0 | 0 | 53 | 89 |
| 0 | 0 | 0 | 36 | 44 |
G:=sub<GL(5,GF(97))| [47,0,0,0,0,0,0,75,0,0,0,22,75,0,0,0,0,0,1,96,0,0,0,1,0],[1,0,0,0,0,0,96,1,0,0,0,96,0,0,0,0,0,0,96,1,0,0,0,96,0],[85,0,0,0,0,0,28,79,0,0,0,10,69,0,0,0,0,0,53,36,0,0,0,89,44] >;
C24.S3 in GAP, Magma, Sage, TeX
C_{24}.S_3 % in TeX
G:=Group("C24.S3"); // GroupNames label
G:=SmallGroup(144,29);
// by ID
G=gap.SmallGroup(144,29);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,12,31,50,964,3461]);
// Polycyclic
G:=Group<a,b,c|a^24=b^3=1,c^2=a^9,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=b^-1>;
// generators/relations
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