metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊14D6, C24⋊14D10, D20.1D6, D15⋊1SD16, Dic6⋊1D10, Dic10⋊1D6, D30.20D4, D12.1D10, C120⋊20C22, C60.139C23, Dic15.25D4, C8⋊14(S3×D5), C3⋊2(D5×SD16), C5⋊2(S3×SD16), C40⋊C2⋊5S3, C24⋊C2⋊5D5, D15⋊Q8⋊8C2, C6.28(D4×D5), C30.9(C2×D4), C15⋊3(C2×SD16), (C8×D15)⋊10C2, C10.28(S3×D4), C20⋊D6.2C2, C30.D4⋊9C2, C20.D6⋊9C2, C15⋊3C8⋊35C22, C2.6(C20⋊D6), C20.68(C22×S3), C12.68(C22×D5), (C5×Dic6)⋊14C22, (C3×D20).25C22, (C4×D15).53C22, (C5×D12).25C22, (C3×Dic10)⋊14C22, C4.112(C2×S3×D5), (C5×C24⋊C2)⋊7C2, (C3×C40⋊C2)⋊9C2, SmallGroup(480,331)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊14D6
G = < a,b,c | a40=b6=c2=1, bab-1=a19, cac=a9, cbc=b-1 >
Subgroups: 908 in 136 conjugacy classes, 40 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C2×SD16, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C3×SD16, S3×D4, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, S3×SD16, C15⋊3C8, C120, D30.C2, C15⋊D4, C15⋊Q8, C3×Dic10, C3×D20, C5×Dic6, C5×D12, C4×D15, C2×S3×D5, D5×SD16, C30.D4, C20.D6, C3×C40⋊C2, C5×C24⋊C2, C8×D15, D15⋊Q8, C20⋊D6, C40⋊14D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C22×S3, C2×SD16, C22×D5, S3×D4, S3×D5, D4×D5, S3×SD16, C2×S3×D5, D5×SD16, C20⋊D6, C40⋊14D6
►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)
(1 101 59)(2 120 60 20 102 78)(3 99 61 39 103 57)(4 118 62 18 104 76)(5 97 63 37 105 55)(6 116 64 16 106 74)(7 95 65 35 107 53)(8 114 66 14 108 72)(9 93 67 33 109 51)(10 112 68 12 110 70)(11 91 69 31 111 49)(13 89 71 29 113 47)(15 87 73 27 115 45)(17 85 75 25 117 43)(19 83 77 23 119 41)(21 81 79)(22 100 80 40 82 58)(24 98 42 38 84 56)(26 96 44 36 86 54)(28 94 46 34 88 52)(30 92 48 32 90 50)
(1 79)(2 48)(3 57)(4 66)(5 75)(6 44)(7 53)(8 62)(9 71)(10 80)(11 49)(12 58)(13 67)(14 76)(15 45)(16 54)(17 63)(18 72)(19 41)(20 50)(21 59)(22 68)(23 77)(24 46)(25 55)(26 64)(27 73)(28 42)(29 51)(30 60)(31 69)(32 78)(33 47)(34 56)(35 65)(36 74)(37 43)(38 52)(39 61)(40 70)(81 101)(82 110)(83 119)(84 88)(85 97)(86 106)(87 115)(89 93)(90 102)(91 111)(92 120)(94 98)(95 107)(96 116)(99 103)(100 112)(104 108)(105 117)(109 113)(114 118)
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), (1,101,59)(2,120,60,20,102,78)(3,99,61,39,103,57)(4,118,62,18,104,76)(5,97,63,37,105,55)(6,116,64,16,106,74)(7,95,65,35,107,53)(8,114,66,14,108,72)(9,93,67,33,109,51)(10,112,68,12,110,70)(11,91,69,31,111,49)(13,89,71,29,113,47)(15,87,73,27,115,45)(17,85,75,25,117,43)(19,83,77,23,119,41)(21,81,79)(22,100,80,40,82,58)(24,98,42,38,84,56)(26,96,44,36,86,54)(28,94,46,34,88,52)(30,92,48,32,90,50), (1,79)(2,48)(3,57)(4,66)(5,75)(6,44)(7,53)(8,62)(9,71)(10,80)(11,49)(12,58)(13,67)(14,76)(15,45)(16,54)(17,63)(18,72)(19,41)(20,50)(21,59)(22,68)(23,77)(24,46)(25,55)(26,64)(27,73)(28,42)(29,51)(30,60)(31,69)(32,78)(33,47)(34,56)(35,65)(36,74)(37,43)(38,52)(39,61)(40,70)(81,101)(82,110)(83,119)(84,88)(85,97)(86,106)(87,115)(89,93)(90,102)(91,111)(92,120)(94,98)(95,107)(96,116)(99,103)(100,112)(104,108)(105,117)(109,113)(114,118)>;
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), (1,101,59)(2,120,60,20,102,78)(3,99,61,39,103,57)(4,118,62,18,104,76)(5,97,63,37,105,55)(6,116,64,16,106,74)(7,95,65,35,107,53)(8,114,66,14,108,72)(9,93,67,33,109,51)(10,112,68,12,110,70)(11,91,69,31,111,49)(13,89,71,29,113,47)(15,87,73,27,115,45)(17,85,75,25,117,43)(19,83,77,23,119,41)(21,81,79)(22,100,80,40,82,58)(24,98,42,38,84,56)(26,96,44,36,86,54)(28,94,46,34,88,52)(30,92,48,32,90,50), (1,79)(2,48)(3,57)(4,66)(5,75)(6,44)(7,53)(8,62)(9,71)(10,80)(11,49)(12,58)(13,67)(14,76)(15,45)(16,54)(17,63)(18,72)(19,41)(20,50)(21,59)(22,68)(23,77)(24,46)(25,55)(26,64)(27,73)(28,42)(29,51)(30,60)(31,69)(32,78)(33,47)(34,56)(35,65)(36,74)(37,43)(38,52)(39,61)(40,70)(81,101)(82,110)(83,119)(84,88)(85,97)(86,106)(87,115)(89,93)(90,102)(91,111)(92,120)(94,98)(95,107)(96,116)(99,103)(100,112)(104,108)(105,117)(109,113)(114,118) );
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)], [(1,101,59),(2,120,60,20,102,78),(3,99,61,39,103,57),(4,118,62,18,104,76),(5,97,63,37,105,55),(6,116,64,16,106,74),(7,95,65,35,107,53),(8,114,66,14,108,72),(9,93,67,33,109,51),(10,112,68,12,110,70),(11,91,69,31,111,49),(13,89,71,29,113,47),(15,87,73,27,115,45),(17,85,75,25,117,43),(19,83,77,23,119,41),(21,81,79),(22,100,80,40,82,58),(24,98,42,38,84,56),(26,96,44,36,86,54),(28,94,46,34,88,52),(30,92,48,32,90,50)], [(1,79),(2,48),(3,57),(4,66),(5,75),(6,44),(7,53),(8,62),(9,71),(10,80),(11,49),(12,58),(13,67),(14,76),(15,45),(16,54),(17,63),(18,72),(19,41),(20,50),(21,59),(22,68),(23,77),(24,46),(25,55),(26,64),(27,73),(28,42),(29,51),(30,60),(31,69),(32,78),(33,47),(34,56),(35,65),(36,74),(37,43),(38,52),(39,61),(40,70),(81,101),(82,110),(83,119),(84,88),(85,97),(86,106),(87,115),(89,93),(90,102),(91,111),(92,120),(94,98),(95,107),(96,116),(99,103),(100,112),(104,108),(105,117),(109,113),(114,118)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 12 | 15 | 15 | 20 | 2 | 2 | 12 | 20 | 30 | 2 | 2 | 2 | 40 | 2 | 2 | 30 | 30 | 2 | 2 | 24 | 24 | 4 | 40 | 4 | 4 | 4 | 4 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | SD16 | D10 | D10 | D10 | S3×D4 | S3×D5 | D4×D5 | S3×SD16 | C2×S3×D5 | D5×SD16 | C20⋊D6 | C40⋊14D6 |
| kernel | C40⋊14D6 | C30.D4 | C20.D6 | C3×C40⋊C2 | C5×C24⋊C2 | C8×D15 | D15⋊Q8 | C20⋊D6 | C40⋊C2 | Dic15 | D30 | C24⋊C2 | C40 | Dic10 | D20 | D15 | C24 | Dic6 | D12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C40⋊14D6 ►in GL6(𝔽241)
| 19 | 222 | 0 | 0 | 0 | 0 |
| 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 52 | 0 | 0 |
| 0 | 0 | 189 | 189 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [19,19,0,0,0,0,222,19,0,0,0,0,0,0,1,189,0,0,0,0,52,189,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,240,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,0,0,0,0,240,1,0,0,0,0,240,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,189,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,240,1] >;
C40⋊14D6 in GAP, Magma, Sage, TeX
C_{40}\rtimes_{14}D_6 % in TeX
G:=Group("C40:14D6"); // GroupNames label
G:=SmallGroup(480,331);
// by ID
G=gap.SmallGroup(480,331);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,135,58,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=c^2=1,b*a*b^-1=a^19,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations