metabelian, supersoluble, monomial, A-group
Aliases: C24.9S3, C32:4C16, C12.7Dic3, C3:(C3:C16), C6.3(C3:C8), (C3xC6).4C8, C8.2(C3:S3), (C3xC24).5C2, (C3xC12).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 >
Quotients: C1, C2, C4, S3, C8, Dic3, C16, C3:S3, C3:C8, C3:Dic3, C3:C16, C32:4C8, C24.S3
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)
(1 138 42)(2 139 43)(3 140 44)(4 141 45)(5 142 46)(6 143 47)(7 144 48)(8 121 25)(9 122 26)(10 123 27)(11 124 28)(12 125 29)(13 126 30)(14 127 31)(15 128 32)(16 129 33)(17 130 34)(18 131 35)(19 132 36)(20 133 37)(21 134 38)(22 135 39)(23 136 40)(24 137 41)(49 94 108)(50 95 109)(51 96 110)(52 73 111)(53 74 112)(54 75 113)(55 76 114)(56 77 115)(57 78 116)(58 79 117)(59 80 118)(60 81 119)(61 82 120)(62 83 97)(63 84 98)(64 85 99)(65 86 100)(66 87 101)(67 88 102)(68 89 103)(69 90 104)(70 91 105)(71 92 106)(72 93 107)
(1 117 10 102 19 111 4 120 13 105 22 114 7 99 16 108)(2 110 11 119 20 104 5 113 14 98 23 107 8 116 17 101)(3 103 12 112 21 97 6 106 15 115 24 100 9 109 18 118)(25 57 34 66 43 51 28 60 37 69 46 54 31 63 40 72)(26 50 35 59 44 68 29 53 38 62 47 71 32 56 41 65)(27 67 36 52 45 61 30 70 39 55 48 64 33 49 42 58)(73 141 82 126 91 135 76 144 85 129 94 138 79 123 88 132)(74 134 83 143 92 128 77 137 86 122 95 131 80 140 89 125)(75 127 84 136 93 121 78 130 87 139 96 124 81 133 90 142)
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), (1,138,42)(2,139,43)(3,140,44)(4,141,45)(5,142,46)(6,143,47)(7,144,48)(8,121,25)(9,122,26)(10,123,27)(11,124,28)(12,125,29)(13,126,30)(14,127,31)(15,128,32)(16,129,33)(17,130,34)(18,131,35)(19,132,36)(20,133,37)(21,134,38)(22,135,39)(23,136,40)(24,137,41)(49,94,108)(50,95,109)(51,96,110)(52,73,111)(53,74,112)(54,75,113)(55,76,114)(56,77,115)(57,78,116)(58,79,117)(59,80,118)(60,81,119)(61,82,120)(62,83,97)(63,84,98)(64,85,99)(65,86,100)(66,87,101)(67,88,102)(68,89,103)(69,90,104)(70,91,105)(71,92,106)(72,93,107), (1,117,10,102,19,111,4,120,13,105,22,114,7,99,16,108)(2,110,11,119,20,104,5,113,14,98,23,107,8,116,17,101)(3,103,12,112,21,97,6,106,15,115,24,100,9,109,18,118)(25,57,34,66,43,51,28,60,37,69,46,54,31,63,40,72)(26,50,35,59,44,68,29,53,38,62,47,71,32,56,41,65)(27,67,36,52,45,61,30,70,39,55,48,64,33,49,42,58)(73,141,82,126,91,135,76,144,85,129,94,138,79,123,88,132)(74,134,83,143,92,128,77,137,86,122,95,131,80,140,89,125)(75,127,84,136,93,121,78,130,87,139,96,124,81,133,90,142)>;
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), (1,138,42)(2,139,43)(3,140,44)(4,141,45)(5,142,46)(6,143,47)(7,144,48)(8,121,25)(9,122,26)(10,123,27)(11,124,28)(12,125,29)(13,126,30)(14,127,31)(15,128,32)(16,129,33)(17,130,34)(18,131,35)(19,132,36)(20,133,37)(21,134,38)(22,135,39)(23,136,40)(24,137,41)(49,94,108)(50,95,109)(51,96,110)(52,73,111)(53,74,112)(54,75,113)(55,76,114)(56,77,115)(57,78,116)(58,79,117)(59,80,118)(60,81,119)(61,82,120)(62,83,97)(63,84,98)(64,85,99)(65,86,100)(66,87,101)(67,88,102)(68,89,103)(69,90,104)(70,91,105)(71,92,106)(72,93,107), (1,117,10,102,19,111,4,120,13,105,22,114,7,99,16,108)(2,110,11,119,20,104,5,113,14,98,23,107,8,116,17,101)(3,103,12,112,21,97,6,106,15,115,24,100,9,109,18,118)(25,57,34,66,43,51,28,60,37,69,46,54,31,63,40,72)(26,50,35,59,44,68,29,53,38,62,47,71,32,56,41,65)(27,67,36,52,45,61,30,70,39,55,48,64,33,49,42,58)(73,141,82,126,91,135,76,144,85,129,94,138,79,123,88,132)(74,134,83,143,92,128,77,137,86,122,95,131,80,140,89,125)(75,127,84,136,93,121,78,130,87,139,96,124,81,133,90,142) );
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)], [(1,138,42),(2,139,43),(3,140,44),(4,141,45),(5,142,46),(6,143,47),(7,144,48),(8,121,25),(9,122,26),(10,123,27),(11,124,28),(12,125,29),(13,126,30),(14,127,31),(15,128,32),(16,129,33),(17,130,34),(18,131,35),(19,132,36),(20,133,37),(21,134,38),(22,135,39),(23,136,40),(24,137,41),(49,94,108),(50,95,109),(51,96,110),(52,73,111),(53,74,112),(54,75,113),(55,76,114),(56,77,115),(57,78,116),(58,79,117),(59,80,118),(60,81,119),(61,82,120),(62,83,97),(63,84,98),(64,85,99),(65,86,100),(66,87,101),(67,88,102),(68,89,103),(69,90,104),(70,91,105),(71,92,106),(72,93,107)], [(1,117,10,102,19,111,4,120,13,105,22,114,7,99,16,108),(2,110,11,119,20,104,5,113,14,98,23,107,8,116,17,101),(3,103,12,112,21,97,6,106,15,115,24,100,9,109,18,118),(25,57,34,66,43,51,28,60,37,69,46,54,31,63,40,72),(26,50,35,59,44,68,29,53,38,62,47,71,32,56,41,65),(27,67,36,52,45,61,30,70,39,55,48,64,33,49,42,58),(73,141,82,126,91,135,76,144,85,129,94,138,79,123,88,132),(74,134,83,143,92,128,77,137,86,122,95,131,80,140,89,125),(75,127,84,136,93,121,78,130,87,139,96,124,81,133,90,142)]])
C24.S3 is a maximal subgroup of
C32:2C32 S3xC3:C16 C24.61D6 C32:2D16 D24.S3 C32:2Q32 C16xC3: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 | C3xC24 | C3xC12 | C3xC6 | 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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