direct product, abelian, monomial
Aliases: C6×C24, SmallGroup(144,104)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C6×C24 |
| C1 — C6×C24 |
| C1 — C6×C24 |
Generators and relations for C6×C24
G = < a,b | a6=b24=1, ab=ba >
Subgroups: 66, all normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, C3×C6, C3×C6, C24, C2×C12, C3×C12, C62, C2×C24, C3×C24, C6×C12, C6×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C32, C12, C2×C6, C2×C8, C3×C6, C24, C2×C12, C3×C12, C62, C2×C24, C3×C24, C6×C12, C6×C24
►Regular action on 144 points
Generators in S144
(1 126 91 41 120 71)(2 127 92 42 97 72)(3 128 93 43 98 49)(4 129 94 44 99 50)(5 130 95 45 100 51)(6 131 96 46 101 52)(7 132 73 47 102 53)(8 133 74 48 103 54)(9 134 75 25 104 55)(10 135 76 26 105 56)(11 136 77 27 106 57)(12 137 78 28 107 58)(13 138 79 29 108 59)(14 139 80 30 109 60)(15 140 81 31 110 61)(16 141 82 32 111 62)(17 142 83 33 112 63)(18 143 84 34 113 64)(19 144 85 35 114 65)(20 121 86 36 115 66)(21 122 87 37 116 67)(22 123 88 38 117 68)(23 124 89 39 118 69)(24 125 90 40 119 70)
(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)
G:=sub<Sym(144)| (1,126,91,41,120,71)(2,127,92,42,97,72)(3,128,93,43,98,49)(4,129,94,44,99,50)(5,130,95,45,100,51)(6,131,96,46,101,52)(7,132,73,47,102,53)(8,133,74,48,103,54)(9,134,75,25,104,55)(10,135,76,26,105,56)(11,136,77,27,106,57)(12,137,78,28,107,58)(13,138,79,29,108,59)(14,139,80,30,109,60)(15,140,81,31,110,61)(16,141,82,32,111,62)(17,142,83,33,112,63)(18,143,84,34,113,64)(19,144,85,35,114,65)(20,121,86,36,115,66)(21,122,87,37,116,67)(22,123,88,38,117,68)(23,124,89,39,118,69)(24,125,90,40,119,70), (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)>;
G:=Group( (1,126,91,41,120,71)(2,127,92,42,97,72)(3,128,93,43,98,49)(4,129,94,44,99,50)(5,130,95,45,100,51)(6,131,96,46,101,52)(7,132,73,47,102,53)(8,133,74,48,103,54)(9,134,75,25,104,55)(10,135,76,26,105,56)(11,136,77,27,106,57)(12,137,78,28,107,58)(13,138,79,29,108,59)(14,139,80,30,109,60)(15,140,81,31,110,61)(16,141,82,32,111,62)(17,142,83,33,112,63)(18,143,84,34,113,64)(19,144,85,35,114,65)(20,121,86,36,115,66)(21,122,87,37,116,67)(22,123,88,38,117,68)(23,124,89,39,118,69)(24,125,90,40,119,70), (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) );
G=PermutationGroup([[(1,126,91,41,120,71),(2,127,92,42,97,72),(3,128,93,43,98,49),(4,129,94,44,99,50),(5,130,95,45,100,51),(6,131,96,46,101,52),(7,132,73,47,102,53),(8,133,74,48,103,54),(9,134,75,25,104,55),(10,135,76,26,105,56),(11,136,77,27,106,57),(12,137,78,28,107,58),(13,138,79,29,108,59),(14,139,80,30,109,60),(15,140,81,31,110,61),(16,141,82,32,111,62),(17,142,83,33,112,63),(18,143,84,34,113,64),(19,144,85,35,114,65),(20,121,86,36,115,66),(21,122,87,37,116,67),(22,123,88,38,117,68),(23,124,89,39,118,69),(24,125,90,40,119,70)], [(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)]])
C6×C24 is a maximal subgroup of
C24.94D6 C12.30Dic6 C24⋊Dic3 C6.4Dic12 C24⋊2Dic3 C24⋊1Dic3 C12.59D12 C12.60D12 C62.84D4 C24.95D6 C24.78D6
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 6A | ··· | 6X | 8A | ··· | 8H | 12A | ··· | 12AF | 24A | ··· | 24BL |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 |
| kernel | C6×C24 | C3×C24 | C6×C12 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 16 | 8 | 8 | 16 | 16 | 64 |
Matrix representation of C6×C24 ►in GL2(𝔽73) generated by
| 65 | 0 |
| 0 | 72 |
| 10 | 0 |
| 0 | 3 |
G:=sub<GL(2,GF(73))| [65,0,0,72],[10,0,0,3] >;
C6×C24 in GAP, Magma, Sage, TeX
C_6\times C_{24} % in TeX
G:=Group("C6xC24"); // GroupNames label
G:=SmallGroup(144,104);
// by ID
G=gap.SmallGroup(144,104);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-2,216,88]);
// Polycyclic
G:=Group<a,b|a^6=b^24=1,a*b=b*a>;
// generators/relations