metabelian, supersoluble, monomial
Aliases: (C6×C12).13C4, (C3×C12).59D4, (C2×C12).96D6, (C6×Q8).19S3, (C2×C12).4Dic3, C12.41(C3⋊D4), (C6×C12).63C22, C62.106(C2×C4), C3⋊2(C12.10D4), C2.7(C62⋊5C4), C12.58D6.6C2, C4.15(C32⋊7D4), C32⋊8(C4.10D4), C6.27(C6.D4), (Q8×C3×C6).4C2, (C2×C4).(C3⋊Dic3), (C2×Q8).4(C3⋊S3), (C2×C6).49(C2×Dic3), C22.4(C2×C3⋊Dic3), (C3×C6).75(C22⋊C4), (C2×C4).4(C2×C3⋊S3), SmallGroup(288,311)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C6×C12).C4
G = < a,b,c | a6=b12=1, c4=b6, ab=ba, cac-1=a-1b6, cbc-1=a3b-1 >
Subgroups: 268 in 114 conjugacy classes, 57 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, C12, C12, C2×C6, M4(2), C2×Q8, C3×C6, C3×C6, C3⋊C8, C2×C12, C3×Q8, C4.10D4, C3×C12, C3×C12, C62, C4.Dic3, C6×Q8, C32⋊4C8, C6×C12, C6×C12, Q8×C32, C12.10D4, C12.58D6, Q8×C3×C6, (C6×C12).C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C3⋊S3, C2×Dic3, C3⋊D4, C4.10D4, C3⋊Dic3, C2×C3⋊S3, C6.D4, C2×C3⋊Dic3, C32⋊7D4, C12.10D4, C62⋊5C4, (C6×C12).C4
►On 144 points
Generators in S144
(1 40 110)(2 41 111)(3 42 112)(4 43 113)(5 44 114)(6 45 115)(7 46 116)(8 47 117)(9 48 118)(10 37 119)(11 38 120)(12 39 109)(13 93 130 19 87 124)(14 94 131 20 88 125)(15 95 132 21 89 126)(16 96 121 22 90 127)(17 85 122 23 91 128)(18 86 123 24 92 129)(25 81 56 31 75 50)(26 82 57 32 76 51)(27 83 58 33 77 52)(28 84 59 34 78 53)(29 73 60 35 79 54)(30 74 49 36 80 55)(61 140 104)(62 141 105)(63 142 106)(64 143 107)(65 144 108)(66 133 97)(67 134 98)(68 135 99)(69 136 100)(70 137 101)(71 138 102)(72 139 103)
(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 13 106 59 7 19 100 53)(2 24 101 52 8 18 107 58)(3 23 108 57 9 17 102 51)(4 22 103 50 10 16 97 56)(5 21 98 55 11 15 104 49)(6 20 105 60 12 14 99 54)(25 113 96 139 31 119 90 133)(26 118 91 138 32 112 85 144)(27 111 86 137 33 117 92 143)(28 116 93 136 34 110 87 142)(29 109 88 135 35 115 94 141)(30 114 95 134 36 120 89 140)(37 121 66 75 43 127 72 81)(38 132 61 80 44 126 67 74)(39 131 68 73 45 125 62 79)(40 130 63 78 46 124 69 84)(41 129 70 83 47 123 64 77)(42 128 65 76 48 122 71 82)
G:=sub<Sym(144)| (1,40,110)(2,41,111)(3,42,112)(4,43,113)(5,44,114)(6,45,115)(7,46,116)(8,47,117)(9,48,118)(10,37,119)(11,38,120)(12,39,109)(13,93,130,19,87,124)(14,94,131,20,88,125)(15,95,132,21,89,126)(16,96,121,22,90,127)(17,85,122,23,91,128)(18,86,123,24,92,129)(25,81,56,31,75,50)(26,82,57,32,76,51)(27,83,58,33,77,52)(28,84,59,34,78,53)(29,73,60,35,79,54)(30,74,49,36,80,55)(61,140,104)(62,141,105)(63,142,106)(64,143,107)(65,144,108)(66,133,97)(67,134,98)(68,135,99)(69,136,100)(70,137,101)(71,138,102)(72,139,103), (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), (1,13,106,59,7,19,100,53)(2,24,101,52,8,18,107,58)(3,23,108,57,9,17,102,51)(4,22,103,50,10,16,97,56)(5,21,98,55,11,15,104,49)(6,20,105,60,12,14,99,54)(25,113,96,139,31,119,90,133)(26,118,91,138,32,112,85,144)(27,111,86,137,33,117,92,143)(28,116,93,136,34,110,87,142)(29,109,88,135,35,115,94,141)(30,114,95,134,36,120,89,140)(37,121,66,75,43,127,72,81)(38,132,61,80,44,126,67,74)(39,131,68,73,45,125,62,79)(40,130,63,78,46,124,69,84)(41,129,70,83,47,123,64,77)(42,128,65,76,48,122,71,82)>;
G:=Group( (1,40,110)(2,41,111)(3,42,112)(4,43,113)(5,44,114)(6,45,115)(7,46,116)(8,47,117)(9,48,118)(10,37,119)(11,38,120)(12,39,109)(13,93,130,19,87,124)(14,94,131,20,88,125)(15,95,132,21,89,126)(16,96,121,22,90,127)(17,85,122,23,91,128)(18,86,123,24,92,129)(25,81,56,31,75,50)(26,82,57,32,76,51)(27,83,58,33,77,52)(28,84,59,34,78,53)(29,73,60,35,79,54)(30,74,49,36,80,55)(61,140,104)(62,141,105)(63,142,106)(64,143,107)(65,144,108)(66,133,97)(67,134,98)(68,135,99)(69,136,100)(70,137,101)(71,138,102)(72,139,103), (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), (1,13,106,59,7,19,100,53)(2,24,101,52,8,18,107,58)(3,23,108,57,9,17,102,51)(4,22,103,50,10,16,97,56)(5,21,98,55,11,15,104,49)(6,20,105,60,12,14,99,54)(25,113,96,139,31,119,90,133)(26,118,91,138,32,112,85,144)(27,111,86,137,33,117,92,143)(28,116,93,136,34,110,87,142)(29,109,88,135,35,115,94,141)(30,114,95,134,36,120,89,140)(37,121,66,75,43,127,72,81)(38,132,61,80,44,126,67,74)(39,131,68,73,45,125,62,79)(40,130,63,78,46,124,69,84)(41,129,70,83,47,123,64,77)(42,128,65,76,48,122,71,82) );
G=PermutationGroup([[(1,40,110),(2,41,111),(3,42,112),(4,43,113),(5,44,114),(6,45,115),(7,46,116),(8,47,117),(9,48,118),(10,37,119),(11,38,120),(12,39,109),(13,93,130,19,87,124),(14,94,131,20,88,125),(15,95,132,21,89,126),(16,96,121,22,90,127),(17,85,122,23,91,128),(18,86,123,24,92,129),(25,81,56,31,75,50),(26,82,57,32,76,51),(27,83,58,33,77,52),(28,84,59,34,78,53),(29,73,60,35,79,54),(30,74,49,36,80,55),(61,140,104),(62,141,105),(63,142,106),(64,143,107),(65,144,108),(66,133,97),(67,134,98),(68,135,99),(69,136,100),(70,137,101),(71,138,102),(72,139,103)], [(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)], [(1,13,106,59,7,19,100,53),(2,24,101,52,8,18,107,58),(3,23,108,57,9,17,102,51),(4,22,103,50,10,16,97,56),(5,21,98,55,11,15,104,49),(6,20,105,60,12,14,99,54),(25,113,96,139,31,119,90,133),(26,118,91,138,32,112,85,144),(27,111,86,137,33,117,92,143),(28,116,93,136,34,110,87,142),(29,109,88,135,35,115,94,141),(30,114,95,134,36,120,89,140),(37,121,66,75,43,127,72,81),(38,132,61,80,44,126,67,74),(39,131,68,73,45,125,62,79),(40,130,63,78,46,124,69,84),(41,129,70,83,47,123,64,77),(42,128,65,76,48,122,71,82)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 36 | 36 | 36 | 36 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C4 | S3 | D4 | Dic3 | D6 | C3⋊D4 | C4.10D4 | C12.10D4 |
| kernel | (C6×C12).C4 | C12.58D6 | Q8×C3×C6 | C6×C12 | C6×Q8 | C3×C12 | C2×C12 | C2×C12 | C12 | C32 | C3 |
| # reps | 1 | 2 | 1 | 4 | 4 | 2 | 8 | 4 | 16 | 1 | 8 |
Matrix representation of (C6×C12).C4 ►in GL6(𝔽73)
| 65 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 | 0 | 0 |
| 0 | 0 | 56 | 31 | 0 | 0 |
| 0 | 0 | 31 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 29 |
| 0 | 0 | 0 | 0 | 29 | 63 |
| 0 | 47 | 0 | 0 | 0 | 0 |
| 14 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
G:=sub<GL(6,GF(73))| [65,0,0,0,0,0,0,9,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[64,0,0,0,0,0,0,65,0,0,0,0,0,0,56,31,0,0,0,0,31,17,0,0,0,0,0,0,10,29,0,0,0,0,29,63],[0,14,0,0,0,0,47,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
(C6×C12).C4 in GAP, Magma, Sage, TeX
(C_6\times C_{12}).C_4 % in TeX
G:=Group("(C6xC12).C4"); // GroupNames label
G:=SmallGroup(288,311);
// by ID
G=gap.SmallGroup(288,311);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,120,219,100,675,2693,9414]);
// Polycyclic
G:=Group<a,b,c|a^6=b^12=1,c^4=b^6,a*b=b*a,c*a*c^-1=a^-1*b^6,c*b*c^-1=a^3*b^-1>;
// generators/relations