direct product, metabelian, supersoluble, monomial
Aliases: C3×C15⋊Q8, C15⋊4Dic6, C30.34D6, C32⋊4Dic10, Dic15.2C6, C15⋊(C3×Q8), (C3×C15)⋊3Q8, C6.7(C6×D5), C5⋊1(C3×Dic6), C10.7(S3×C6), C30.7(C2×C6), C6.34(S3×D5), Dic3.(C3×D5), C3⋊1(C3×Dic10), (C3×C6).19D10, (C3×C30).7C22, (C3×Dic3).2D5, (C3×Dic5).6S3, (C3×Dic5).1C6, (C5×Dic3).1C6, Dic5.1(C3×S3), (Dic3×C15).2C2, (C3×Dic15).2C2, (C32×Dic5).1C2, C2.7(C3×S3×D5), SmallGroup(360,64)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C15⋊Q8
G = < a,b,c,d | a3=b15=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b11, dbd-1=b4, dcd-1=c-1 >
Subgroups: 172 in 54 conjugacy classes, 28 normal (all characteristic)
C1, C2, C3, C3, C4, C5, C6, C6, Q8, C32, C10, Dic3, Dic3, C12, C15, C15, C3×C6, Dic5, Dic5, C20, Dic6, C3×Q8, C30, C30, C3×Dic3, C3×Dic3, C3×C12, Dic10, C3×C15, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C3×Dic6, C3×C30, C15⋊Q8, C3×Dic10, C32×Dic5, Dic3×C15, C3×Dic15, C3×C15⋊Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, D5, D6, C2×C6, C3×S3, D10, Dic6, C3×Q8, C3×D5, S3×C6, Dic10, S3×D5, C6×D5, C3×Dic6, C15⋊Q8, C3×Dic10, C3×S3×D5, C3×C15⋊Q8
►On 120 points
Generators in S120
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(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 59 21 31)(2 55 22 42)(3 51 23 38)(4 47 24 34)(5 58 25 45)(6 54 26 41)(7 50 27 37)(8 46 28 33)(9 57 29 44)(10 53 30 40)(11 49 16 36)(12 60 17 32)(13 56 18 43)(14 52 19 39)(15 48 20 35)(61 93 78 119)(62 104 79 115)(63 100 80 111)(64 96 81 107)(65 92 82 118)(66 103 83 114)(67 99 84 110)(68 95 85 106)(69 91 86 117)(70 102 87 113)(71 98 88 109)(72 94 89 120)(73 105 90 116)(74 101 76 112)(75 97 77 108)
(1 83 21 66)(2 87 22 70)(3 76 23 74)(4 80 24 63)(5 84 25 67)(6 88 26 71)(7 77 27 75)(8 81 28 64)(9 85 29 68)(10 89 30 72)(11 78 16 61)(12 82 17 65)(13 86 18 69)(14 90 19 73)(15 79 20 62)(31 114 59 103)(32 118 60 92)(33 107 46 96)(34 111 47 100)(35 115 48 104)(36 119 49 93)(37 108 50 97)(38 112 51 101)(39 116 52 105)(40 120 53 94)(41 109 54 98)(42 113 55 102)(43 117 56 91)(44 106 57 95)(45 110 58 99)
G:=sub<Sym(120)| (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,51,56)(47,52,57)(48,53,58)(49,54,59)(50,55,60)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,86,81)(77,87,82)(78,88,83)(79,89,84)(80,90,85)(91,96,101)(92,97,102)(93,98,103)(94,99,104)(95,100,105)(106,111,116)(107,112,117)(108,113,118)(109,114,119)(110,115,120), (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), (1,59,21,31)(2,55,22,42)(3,51,23,38)(4,47,24,34)(5,58,25,45)(6,54,26,41)(7,50,27,37)(8,46,28,33)(9,57,29,44)(10,53,30,40)(11,49,16,36)(12,60,17,32)(13,56,18,43)(14,52,19,39)(15,48,20,35)(61,93,78,119)(62,104,79,115)(63,100,80,111)(64,96,81,107)(65,92,82,118)(66,103,83,114)(67,99,84,110)(68,95,85,106)(69,91,86,117)(70,102,87,113)(71,98,88,109)(72,94,89,120)(73,105,90,116)(74,101,76,112)(75,97,77,108), (1,83,21,66)(2,87,22,70)(3,76,23,74)(4,80,24,63)(5,84,25,67)(6,88,26,71)(7,77,27,75)(8,81,28,64)(9,85,29,68)(10,89,30,72)(11,78,16,61)(12,82,17,65)(13,86,18,69)(14,90,19,73)(15,79,20,62)(31,114,59,103)(32,118,60,92)(33,107,46,96)(34,111,47,100)(35,115,48,104)(36,119,49,93)(37,108,50,97)(38,112,51,101)(39,116,52,105)(40,120,53,94)(41,109,54,98)(42,113,55,102)(43,117,56,91)(44,106,57,95)(45,110,58,99)>;
G:=Group( (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,51,56)(47,52,57)(48,53,58)(49,54,59)(50,55,60)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,86,81)(77,87,82)(78,88,83)(79,89,84)(80,90,85)(91,96,101)(92,97,102)(93,98,103)(94,99,104)(95,100,105)(106,111,116)(107,112,117)(108,113,118)(109,114,119)(110,115,120), (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), (1,59,21,31)(2,55,22,42)(3,51,23,38)(4,47,24,34)(5,58,25,45)(6,54,26,41)(7,50,27,37)(8,46,28,33)(9,57,29,44)(10,53,30,40)(11,49,16,36)(12,60,17,32)(13,56,18,43)(14,52,19,39)(15,48,20,35)(61,93,78,119)(62,104,79,115)(63,100,80,111)(64,96,81,107)(65,92,82,118)(66,103,83,114)(67,99,84,110)(68,95,85,106)(69,91,86,117)(70,102,87,113)(71,98,88,109)(72,94,89,120)(73,105,90,116)(74,101,76,112)(75,97,77,108), (1,83,21,66)(2,87,22,70)(3,76,23,74)(4,80,24,63)(5,84,25,67)(6,88,26,71)(7,77,27,75)(8,81,28,64)(9,85,29,68)(10,89,30,72)(11,78,16,61)(12,82,17,65)(13,86,18,69)(14,90,19,73)(15,79,20,62)(31,114,59,103)(32,118,60,92)(33,107,46,96)(34,111,47,100)(35,115,48,104)(36,119,49,93)(37,108,50,97)(38,112,51,101)(39,116,52,105)(40,120,53,94)(41,109,54,98)(42,113,55,102)(43,117,56,91)(44,106,57,95)(45,110,58,99) );
G=PermutationGroup([[(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,36,41),(32,37,42),(33,38,43),(34,39,44),(35,40,45),(46,51,56),(47,52,57),(48,53,58),(49,54,59),(50,55,60),(61,71,66),(62,72,67),(63,73,68),(64,74,69),(65,75,70),(76,86,81),(77,87,82),(78,88,83),(79,89,84),(80,90,85),(91,96,101),(92,97,102),(93,98,103),(94,99,104),(95,100,105),(106,111,116),(107,112,117),(108,113,118),(109,114,119),(110,115,120)], [(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)], [(1,59,21,31),(2,55,22,42),(3,51,23,38),(4,47,24,34),(5,58,25,45),(6,54,26,41),(7,50,27,37),(8,46,28,33),(9,57,29,44),(10,53,30,40),(11,49,16,36),(12,60,17,32),(13,56,18,43),(14,52,19,39),(15,48,20,35),(61,93,78,119),(62,104,79,115),(63,100,80,111),(64,96,81,107),(65,92,82,118),(66,103,83,114),(67,99,84,110),(68,95,85,106),(69,91,86,117),(70,102,87,113),(71,98,88,109),(72,94,89,120),(73,105,90,116),(74,101,76,112),(75,97,77,108)], [(1,83,21,66),(2,87,22,70),(3,76,23,74),(4,80,24,63),(5,84,25,67),(6,88,26,71),(7,77,27,75),(8,81,28,64),(9,85,29,68),(10,89,30,72),(11,78,16,61),(12,82,17,65),(13,86,18,69),(14,90,19,73),(15,79,20,62),(31,114,59,103),(32,118,60,92),(33,107,46,96),(34,111,47,100),(35,115,48,104),(36,119,49,93),(37,108,50,97),(38,112,51,101),(39,116,52,105),(40,120,53,94),(41,109,54,98),(42,113,55,102),(43,117,56,91),(44,106,57,95),(45,110,58,99)]])
63 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | 10B | 12A | 12B | 12C | ··· | 12J | 12K | 12L | 15A | 15B | 15C | 15D | 15E | ··· | 15J | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 30E | ··· | 30J | 60A | ··· | 60H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 10 | 30 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 10 | ··· | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | - | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | Q8 | D5 | D6 | C3×S3 | D10 | Dic6 | C3×Q8 | C3×D5 | S3×C6 | Dic10 | C6×D5 | C3×Dic6 | C3×Dic10 | S3×D5 | C15⋊Q8 | C3×S3×D5 | C3×C15⋊Q8 |
| kernel | C3×C15⋊Q8 | C32×Dic5 | Dic3×C15 | C3×Dic15 | C15⋊Q8 | C5×Dic3 | C3×Dic5 | Dic15 | C3×Dic5 | C3×C15 | C3×Dic3 | C30 | Dic5 | C3×C6 | C15 | C15 | Dic3 | C10 | C32 | C6 | C5 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C3×C15⋊Q8 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 60 | 44 | 0 | 0 |
| 17 | 44 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 25 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 5 |
| 0 | 0 | 37 | 27 |
| 8 | 20 | 0 | 0 |
| 6 | 53 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 9 | 50 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[60,17,0,0,44,44,0,0,0,0,47,25,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,34,37,0,0,5,27],[8,6,0,0,20,53,0,0,0,0,11,9,0,0,0,50] >;
C3×C15⋊Q8 in GAP, Magma, Sage, TeX
C_3\times C_{15}\rtimes Q_8 % in TeX
G:=Group("C3xC15:Q8"); // GroupNames label
G:=SmallGroup(360,64);
// by ID
G=gap.SmallGroup(360,64);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,169,79,730,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^15=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^11,d*b*d^-1=b^4,d*c*d^-1=c^-1>;
// generators/relations