metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊17D6, C24⋊20D10, C120⋊24C22, D15⋊3M4(2), C60.169C23, C3⋊C8⋊20D10, C8⋊15(S3×D5), C8⋊D5⋊5S3, C8⋊S3⋊5D5, C5⋊2C8⋊20D6, D6.6(C4×D5), C5⋊6(S3×M4(2)), C3⋊2(D5×M4(2)), (C8×D15)⋊13C2, (C4×D5).55D6, D15⋊2C8⋊8C2, (C4×S3).31D10, D10.17(C4×S3), D30.27(C2×C4), D30.C2.3C4, C15⋊10(C2×M4(2)), (S3×Dic5).3C4, (D5×Dic3).3C4, C15⋊3C8⋊41C22, D6.Dic5⋊10C2, C30.35(C22×C4), Dic3.10(C4×D5), Dic5.22(C4×S3), C20.32D6⋊10C2, (S3×C20).31C22, C20.166(C22×S3), Dic15.34(C2×C4), (C4×D15).61C22, (D5×C12).55C22, C12.166(C22×D5), C6.4(C2×C4×D5), C2.7(C4×S3×D5), (C2×S3×D5).3C4, (C4×S3×D5).8C2, C10.35(S3×C2×C4), C4.139(C2×S3×D5), (C5×C8⋊S3)⋊6C2, (C3×C8⋊D5)⋊8C2, (C5×C3⋊C8)⋊20C22, (C6×D5).2(C2×C4), (S3×C10).17(C2×C4), (C3×C5⋊2C8)⋊20C22, (C3×Dic5).2(C2×C4), (C5×Dic3).18(C2×C4), SmallGroup(480,322)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊D6
G = < a,b,c | a40=b6=c2=1, bab-1=a29, cac=a9, cbc=b-1 >
Subgroups: 636 in 136 conjugacy classes, 52 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, C24, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×M4(2), C5⋊2C8, C5⋊2C8, C40, C40, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C8, C8⋊S3, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C8×D5, C8⋊D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×C4×D5, S3×M4(2), C5×C3⋊C8, C3×C5⋊2C8, C15⋊3C8, C120, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, D5×M4(2), D15⋊2C8, C20.32D6, D6.Dic5, C3×C8⋊D5, C5×C8⋊S3, C8×D15, C4×S3×D5, C40⋊D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, D10, C4×S3, C22×S3, C2×M4(2), C4×D5, C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, S3×M4(2), C2×S3×D5, D5×M4(2), C4×S3×D5, C40⋊D6
►On 120 points
(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)
(1 49 120)(2 78 81 30 50 109)(3 67 82 19 51 98)(4 56 83 8 52 87)(5 45 84 37 53 116)(6 74 85 26 54 105)(7 63 86 15 55 94)(9 41 88 33 57 112)(10 70 89 22 58 101)(11 59 90)(12 48 91 40 60 119)(13 77 92 29 61 108)(14 66 93 18 62 97)(16 44 95 36 64 115)(17 73 96 25 65 104)(20 80 99 32 68 111)(21 69 100)(23 47 102 39 71 118)(24 76 103 28 72 107)(27 43 106 35 75 114)(31 79 110)(34 46 113 38 42 117)
(1 100)(2 109)(3 118)(4 87)(5 96)(6 105)(7 114)(8 83)(9 92)(10 101)(11 110)(12 119)(13 88)(14 97)(15 106)(16 115)(17 84)(18 93)(19 102)(20 111)(21 120)(22 89)(23 98)(24 107)(25 116)(26 85)(27 94)(28 103)(29 112)(30 81)(31 90)(32 99)(33 108)(34 117)(35 86)(36 95)(37 104)(38 113)(39 82)(40 91)(41 77)(42 46)(43 55)(44 64)(45 73)(47 51)(48 60)(49 69)(50 78)(52 56)(53 65)(54 74)(57 61)(58 70)(59 79)(62 66)(63 75)(67 71)(68 80)(72 76)
G:=sub<Sym(120)| (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), (1,49,120)(2,78,81,30,50,109)(3,67,82,19,51,98)(4,56,83,8,52,87)(5,45,84,37,53,116)(6,74,85,26,54,105)(7,63,86,15,55,94)(9,41,88,33,57,112)(10,70,89,22,58,101)(11,59,90)(12,48,91,40,60,119)(13,77,92,29,61,108)(14,66,93,18,62,97)(16,44,95,36,64,115)(17,73,96,25,65,104)(20,80,99,32,68,111)(21,69,100)(23,47,102,39,71,118)(24,76,103,28,72,107)(27,43,106,35,75,114)(31,79,110)(34,46,113,38,42,117), (1,100)(2,109)(3,118)(4,87)(5,96)(6,105)(7,114)(8,83)(9,92)(10,101)(11,110)(12,119)(13,88)(14,97)(15,106)(16,115)(17,84)(18,93)(19,102)(20,111)(21,120)(22,89)(23,98)(24,107)(25,116)(26,85)(27,94)(28,103)(29,112)(30,81)(31,90)(32,99)(33,108)(34,117)(35,86)(36,95)(37,104)(38,113)(39,82)(40,91)(41,77)(42,46)(43,55)(44,64)(45,73)(47,51)(48,60)(49,69)(50,78)(52,56)(53,65)(54,74)(57,61)(58,70)(59,79)(62,66)(63,75)(67,71)(68,80)(72,76)>;
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), (1,49,120)(2,78,81,30,50,109)(3,67,82,19,51,98)(4,56,83,8,52,87)(5,45,84,37,53,116)(6,74,85,26,54,105)(7,63,86,15,55,94)(9,41,88,33,57,112)(10,70,89,22,58,101)(11,59,90)(12,48,91,40,60,119)(13,77,92,29,61,108)(14,66,93,18,62,97)(16,44,95,36,64,115)(17,73,96,25,65,104)(20,80,99,32,68,111)(21,69,100)(23,47,102,39,71,118)(24,76,103,28,72,107)(27,43,106,35,75,114)(31,79,110)(34,46,113,38,42,117), (1,100)(2,109)(3,118)(4,87)(5,96)(6,105)(7,114)(8,83)(9,92)(10,101)(11,110)(12,119)(13,88)(14,97)(15,106)(16,115)(17,84)(18,93)(19,102)(20,111)(21,120)(22,89)(23,98)(24,107)(25,116)(26,85)(27,94)(28,103)(29,112)(30,81)(31,90)(32,99)(33,108)(34,117)(35,86)(36,95)(37,104)(38,113)(39,82)(40,91)(41,77)(42,46)(43,55)(44,64)(45,73)(47,51)(48,60)(49,69)(50,78)(52,56)(53,65)(54,74)(57,61)(58,70)(59,79)(62,66)(63,75)(67,71)(68,80)(72,76) );
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)], [(1,49,120),(2,78,81,30,50,109),(3,67,82,19,51,98),(4,56,83,8,52,87),(5,45,84,37,53,116),(6,74,85,26,54,105),(7,63,86,15,55,94),(9,41,88,33,57,112),(10,70,89,22,58,101),(11,59,90),(12,48,91,40,60,119),(13,77,92,29,61,108),(14,66,93,18,62,97),(16,44,95,36,64,115),(17,73,96,25,65,104),(20,80,99,32,68,111),(21,69,100),(23,47,102,39,71,118),(24,76,103,28,72,107),(27,43,106,35,75,114),(31,79,110),(34,46,113,38,42,117)], [(1,100),(2,109),(3,118),(4,87),(5,96),(6,105),(7,114),(8,83),(9,92),(10,101),(11,110),(12,119),(13,88),(14,97),(15,106),(16,115),(17,84),(18,93),(19,102),(20,111),(21,120),(22,89),(23,98),(24,107),(25,116),(26,85),(27,94),(28,103),(29,112),(30,81),(31,90),(32,99),(33,108),(34,117),(35,86),(36,95),(37,104),(38,113),(39,82),(40,91),(41,77),(42,46),(43,55),(44,64),(45,73),(47,51),(48,60),(49,69),(50,78),(52,56),(53,65),(54,74),(57,61),(58,70),(59,79),(62,66),(63,75),(67,71),(68,80),(72,76)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 40A | 40B | 40C | 40D | 40E | 40F | 40G | 40H | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 6 | 10 | 15 | 15 | 2 | 1 | 1 | 6 | 10 | 15 | 15 | 2 | 2 | 2 | 20 | 2 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 12 | 12 | 2 | 2 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 12 | 12 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | D6 | M4(2) | D10 | D10 | D10 | C4×S3 | C4×S3 | C4×D5 | C4×D5 | S3×D5 | S3×M4(2) | C2×S3×D5 | D5×M4(2) | C4×S3×D5 | C40⋊D6 |
| kernel | C40⋊D6 | D15⋊2C8 | C20.32D6 | D6.Dic5 | C3×C8⋊D5 | C5×C8⋊S3 | C8×D15 | C4×S3×D5 | D5×Dic3 | S3×Dic5 | D30.C2 | C2×S3×D5 | C8⋊D5 | C8⋊S3 | C5⋊2C8 | C40 | C4×D5 | D15 | C3⋊C8 | C24 | C4×S3 | Dic5 | D10 | Dic3 | D6 | C8 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C40⋊D6 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 51 | 0 | 0 |
| 0 | 0 | 190 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 177 |
| 0 | 0 | 0 | 0 | 178 | 1 |
| 239 | 49 | 0 | 0 | 0 | 0 |
| 177 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 51 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 128 | 240 |
| 239 | 49 | 0 | 0 | 0 | 0 |
| 177 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 51 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,190,0,0,0,0,51,190,0,0,0,0,0,0,240,178,0,0,0,0,177,1],[239,177,0,0,0,0,49,1,0,0,0,0,0,0,1,51,0,0,0,0,0,240,0,0,0,0,0,0,1,128,0,0,0,0,0,240],[239,177,0,0,0,0,49,2,0,0,0,0,0,0,1,51,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;
C40⋊D6 in GAP, Magma, Sage, TeX
C_{40}\rtimes D_6 % in TeX
G:=Group("C40:D6"); // GroupNames label
G:=SmallGroup(480,322);
// by ID
G=gap.SmallGroup(480,322);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,219,58,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=c^2=1,b*a*b^-1=a^29,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations