direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D10.3Q8, C30.17C42, (C2×F5)⋊C12, (C2×C60)⋊5C4, (C6×F5)⋊3C4, (C2×C12)⋊3F5, (C2×C20)⋊2C12, C6.17(C4×F5), C2.5(C12×F5), C10.5(C4×C12), (C6×D5).79D4, C6.21(C4⋊F5), C30.21(C4⋊C4), (C6×D5).13Q8, D10.3(C3×Q8), (C6×Dic5)⋊16C4, (C2×Dic5)⋊6C12, D10.17(C3×D4), D10.7(C2×C12), (C22×F5).1C6, C22.13(C6×F5), C6.30(C22⋊F5), C30.30(C22⋊C4), C15⋊3(C2.C42), D5.(C3×C4⋊C4), (C2×C4)⋊2(C3×F5), (C2×C4×D5).8C6, C2.3(C3×C4⋊F5), (C2×C6×F5).4C2, C10.7(C3×C4⋊C4), C5⋊(C3×C2.C42), D5.(C3×C22⋊C4), (D5×C2×C12).23C2, (C2×C6).56(C2×F5), (C2×C30).52(C2×C4), (C2×C10).9(C2×C12), C2.2(C3×C22⋊F5), (C3×D5).3(C4⋊C4), (C6×D5).45(C2×C4), C10.4(C3×C22⋊C4), (D5×C2×C6).148C22, (C3×D5).3(C22⋊C4), (C22×D5).37(C2×C6), SmallGroup(480,286)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D10.3Q8
G = < a,b,c,d,e | a3=b10=c2=d4=1, e2=b4cd2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b2c, ede-1=b5d-1 >
Subgroups: 536 in 152 conjugacy classes, 64 normal (36 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, C12, C2×C6, C2×C6, C15, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C30, C2.C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C22×C12, C3×Dic5, C60, C3×F5, C6×D5, C6×D5, C2×C30, C2×C4×D5, C22×F5, C3×C2.C42, D5×C12, C6×Dic5, C2×C60, C6×F5, C6×F5, D5×C2×C6, D10.3Q8, D5×C2×C12, C2×C6×F5, C3×D10.3Q8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, F5, C2×C12, C3×D4, C3×Q8, C2.C42, C2×F5, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×F5, C4×F5, C4⋊F5, C22⋊F5, C3×C2.C42, C6×F5, D10.3Q8, C12×F5, C3×C4⋊F5, C3×C22⋊F5, C3×D10.3Q8
►On 120 points
Generators in S120
(1 70 50)(2 61 41)(3 62 42)(4 63 43)(5 64 44)(6 65 45)(7 66 46)(8 67 47)(9 68 48)(10 69 49)(11 105 85)(12 106 86)(13 107 87)(14 108 88)(15 109 89)(16 110 90)(17 101 81)(18 102 82)(19 103 83)(20 104 84)(21 57 37)(22 58 38)(23 59 39)(24 60 40)(25 51 31)(26 52 32)(27 53 33)(28 54 34)(29 55 35)(30 56 36)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 30)(7 29)(8 28)(9 27)(10 26)(11 116)(12 115)(13 114)(14 113)(15 112)(16 111)(17 120)(18 119)(19 118)(20 117)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)
(1 77 21 90)(2 78 22 81)(3 79 23 82)(4 80 24 83)(5 71 25 84)(6 72 26 85)(7 73 27 86)(8 74 28 87)(9 75 29 88)(10 76 30 89)(11 65 112 52)(12 66 113 53)(13 67 114 54)(14 68 115 55)(15 69 116 56)(16 70 117 57)(17 61 118 58)(18 62 119 59)(19 63 120 60)(20 64 111 51)(31 104 44 91)(32 105 45 92)(33 106 46 93)(34 107 47 94)(35 108 48 95)(36 109 49 96)(37 110 50 97)(38 101 41 98)(39 102 42 99)(40 103 43 100)
(2 8 10 4)(3 5 9 7)(11 117)(12 114 20 120)(13 111 19 113)(14 118 18 116)(15 115 17 119)(16 112)(22 28 30 24)(23 25 29 27)(31 35 33 39)(34 36 40 38)(41 47 49 43)(42 44 48 46)(51 55 53 59)(54 56 60 58)(61 67 69 63)(62 64 68 66)(71 83 73 87)(72 90)(74 84 80 86)(75 81 79 89)(76 88 78 82)(77 85)(91 103 93 107)(92 110)(94 104 100 106)(95 101 99 109)(96 108 98 102)(97 105)
G:=sub<Sym(120)| (1,70,50)(2,61,41)(3,62,42)(4,63,43)(5,64,44)(6,65,45)(7,66,46)(8,67,47)(9,68,48)(10,69,49)(11,105,85)(12,106,86)(13,107,87)(14,108,88)(15,109,89)(16,110,90)(17,101,81)(18,102,82)(19,103,83)(20,104,84)(21,57,37)(22,58,38)(23,59,39)(24,60,40)(25,51,31)(26,52,32)(27,53,33)(28,54,34)(29,55,35)(30,56,36)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,30)(7,29)(8,28)(9,27)(10,26)(11,116)(12,115)(13,114)(14,113)(15,112)(16,111)(17,120)(18,119)(19,118)(20,117)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101), (1,77,21,90)(2,78,22,81)(3,79,23,82)(4,80,24,83)(5,71,25,84)(6,72,26,85)(7,73,27,86)(8,74,28,87)(9,75,29,88)(10,76,30,89)(11,65,112,52)(12,66,113,53)(13,67,114,54)(14,68,115,55)(15,69,116,56)(16,70,117,57)(17,61,118,58)(18,62,119,59)(19,63,120,60)(20,64,111,51)(31,104,44,91)(32,105,45,92)(33,106,46,93)(34,107,47,94)(35,108,48,95)(36,109,49,96)(37,110,50,97)(38,101,41,98)(39,102,42,99)(40,103,43,100), (2,8,10,4)(3,5,9,7)(11,117)(12,114,20,120)(13,111,19,113)(14,118,18,116)(15,115,17,119)(16,112)(22,28,30,24)(23,25,29,27)(31,35,33,39)(34,36,40,38)(41,47,49,43)(42,44,48,46)(51,55,53,59)(54,56,60,58)(61,67,69,63)(62,64,68,66)(71,83,73,87)(72,90)(74,84,80,86)(75,81,79,89)(76,88,78,82)(77,85)(91,103,93,107)(92,110)(94,104,100,106)(95,101,99,109)(96,108,98,102)(97,105)>;
G:=Group( (1,70,50)(2,61,41)(3,62,42)(4,63,43)(5,64,44)(6,65,45)(7,66,46)(8,67,47)(9,68,48)(10,69,49)(11,105,85)(12,106,86)(13,107,87)(14,108,88)(15,109,89)(16,110,90)(17,101,81)(18,102,82)(19,103,83)(20,104,84)(21,57,37)(22,58,38)(23,59,39)(24,60,40)(25,51,31)(26,52,32)(27,53,33)(28,54,34)(29,55,35)(30,56,36)(71,111,91)(72,112,92)(73,113,93)(74,114,94)(75,115,95)(76,116,96)(77,117,97)(78,118,98)(79,119,99)(80,120,100), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,30)(7,29)(8,28)(9,27)(10,26)(11,116)(12,115)(13,114)(14,113)(15,112)(16,111)(17,120)(18,119)(19,118)(20,117)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101), (1,77,21,90)(2,78,22,81)(3,79,23,82)(4,80,24,83)(5,71,25,84)(6,72,26,85)(7,73,27,86)(8,74,28,87)(9,75,29,88)(10,76,30,89)(11,65,112,52)(12,66,113,53)(13,67,114,54)(14,68,115,55)(15,69,116,56)(16,70,117,57)(17,61,118,58)(18,62,119,59)(19,63,120,60)(20,64,111,51)(31,104,44,91)(32,105,45,92)(33,106,46,93)(34,107,47,94)(35,108,48,95)(36,109,49,96)(37,110,50,97)(38,101,41,98)(39,102,42,99)(40,103,43,100), (2,8,10,4)(3,5,9,7)(11,117)(12,114,20,120)(13,111,19,113)(14,118,18,116)(15,115,17,119)(16,112)(22,28,30,24)(23,25,29,27)(31,35,33,39)(34,36,40,38)(41,47,49,43)(42,44,48,46)(51,55,53,59)(54,56,60,58)(61,67,69,63)(62,64,68,66)(71,83,73,87)(72,90)(74,84,80,86)(75,81,79,89)(76,88,78,82)(77,85)(91,103,93,107)(92,110)(94,104,100,106)(95,101,99,109)(96,108,98,102)(97,105) );
G=PermutationGroup([[(1,70,50),(2,61,41),(3,62,42),(4,63,43),(5,64,44),(6,65,45),(7,66,46),(8,67,47),(9,68,48),(10,69,49),(11,105,85),(12,106,86),(13,107,87),(14,108,88),(15,109,89),(16,110,90),(17,101,81),(18,102,82),(19,103,83),(20,104,84),(21,57,37),(22,58,38),(23,59,39),(24,60,40),(25,51,31),(26,52,32),(27,53,33),(28,54,34),(29,55,35),(30,56,36),(71,111,91),(72,112,92),(73,113,93),(74,114,94),(75,115,95),(76,116,96),(77,117,97),(78,118,98),(79,119,99),(80,120,100)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,30),(7,29),(8,28),(9,27),(10,26),(11,116),(12,115),(13,114),(14,113),(15,112),(16,111),(17,120),(18,119),(19,118),(20,117),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(71,90),(72,89),(73,88),(74,87),(75,86),(76,85),(77,84),(78,83),(79,82),(80,81),(91,110),(92,109),(93,108),(94,107),(95,106),(96,105),(97,104),(98,103),(99,102),(100,101)], [(1,77,21,90),(2,78,22,81),(3,79,23,82),(4,80,24,83),(5,71,25,84),(6,72,26,85),(7,73,27,86),(8,74,28,87),(9,75,29,88),(10,76,30,89),(11,65,112,52),(12,66,113,53),(13,67,114,54),(14,68,115,55),(15,69,116,56),(16,70,117,57),(17,61,118,58),(18,62,119,59),(19,63,120,60),(20,64,111,51),(31,104,44,91),(32,105,45,92),(33,106,46,93),(34,107,47,94),(35,108,48,95),(36,109,49,96),(37,110,50,97),(38,101,41,98),(39,102,42,99),(40,103,43,100)], [(2,8,10,4),(3,5,9,7),(11,117),(12,114,20,120),(13,111,19,113),(14,118,18,116),(15,115,17,119),(16,112),(22,28,30,24),(23,25,29,27),(31,35,33,39),(34,36,40,38),(41,47,49,43),(42,44,48,46),(51,55,53,59),(54,56,60,58),(61,67,69,63),(62,64,68,66),(71,83,73,87),(72,90),(74,84,80,86),(75,81,79,89),(76,88,78,82),(77,85),(91,103,93,107),(92,110),(94,104,100,106),(95,101,99,109),(96,108,98,102),(97,105)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | ··· | 4L | 5 | 6A | ··· | 6F | 6G | ··· | 6N | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | ··· | 12X | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C12 | C12 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | F5 | C2×F5 | C3×F5 | C4×F5 | C4⋊F5 | C22⋊F5 | C6×F5 | C12×F5 | C3×C4⋊F5 | C3×C22⋊F5 |
| kernel | C3×D10.3Q8 | D5×C2×C12 | C2×C6×F5 | D10.3Q8 | C6×Dic5 | C2×C60 | C6×F5 | C2×C4×D5 | C22×F5 | C2×Dic5 | C2×C20 | C2×F5 | C6×D5 | C6×D5 | D10 | D10 | C2×C12 | C2×C6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 2 | 4 | 4 | 4 | 16 | 3 | 1 | 6 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C3×D10.3Q8 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 60 | 0 |
| 0 | 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 1 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 42 | 5 | 0 | 0 | 0 | 0 |
| 13 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 14 | 0 | 47 |
| 0 | 0 | 0 | 7 | 14 | 47 |
| 0 | 0 | 47 | 14 | 7 | 0 |
| 0 | 0 | 47 | 0 | 14 | 54 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 32 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 1 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,1,1,1,0,0,60,0,0,0,0,0,0,60,0,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,60,60,60,60],[42,13,0,0,0,0,5,19,0,0,0,0,0,0,54,0,47,47,0,0,14,7,14,0,0,0,0,14,7,14,0,0,47,47,0,54],[1,32,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,60,0,0,0,1,0,0] >;
C3×D10.3Q8 in GAP, Magma, Sage, TeX
C_3\times D_{10}._3Q_8 % in TeX
G:=Group("C3xD10.3Q8"); // GroupNames label
G:=SmallGroup(480,286);
// by ID
G=gap.SmallGroup(480,286);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,176,9414,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^10=c^2=d^4=1,e^2=b^4*c*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^3,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=b^5*d^-1>;
// generators/relations