direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×C4.D4, C23.C60, C60.243D4, M4(2)⋊3C30, C4.9(D4×C15), (C6×D4).8C10, (C2×D4).2C30, (D4×C10).8C6, C12.58(C5×D4), C20.58(C3×D4), (D4×C30).20C2, (C5×M4(2))⋊9C6, C22.3(C2×C60), (C22×C30).1C4, (C22×C6).1C20, (C3×M4(2))⋊9C10, (C22×C10).3C12, (C15×M4(2))⋊21C2, (C2×C60).418C22, C30.126(C22⋊C4), (C2×C4).1(C2×C30), (C2×C20).59(C2×C6), (C2×C6).20(C2×C20), C2.4(C15×C22⋊C4), C6.22(C5×C22⋊C4), (C2×C30).165(C2×C4), (C2×C10).40(C2×C12), (C2×C12).59(C2×C10), C10.33(C3×C22⋊C4), SmallGroup(480,203)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C4.D4
G = < a,b,c,d | a15=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >
Subgroups: 168 in 92 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), C2×D4, C20, C2×C10, C2×C10, C24, C2×C12, C3×D4, C22×C6, C30, C30, C4.D4, C40, C2×C20, C5×D4, C22×C10, C3×M4(2), C6×D4, C60, C2×C30, C2×C30, C5×M4(2), D4×C10, C3×C4.D4, C120, C2×C60, D4×C15, C22×C30, C5×C4.D4, C15×M4(2), D4×C30, C15×C4.D4
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, C10, C12, C2×C6, C15, C22⋊C4, C20, C2×C10, C2×C12, C3×D4, C30, C4.D4, C2×C20, C5×D4, C3×C22⋊C4, C60, C2×C30, C5×C22⋊C4, C3×C4.D4, C2×C60, D4×C15, C5×C4.D4, C15×C22⋊C4, C15×C4.D4
►On 120 points
Generators in S120
(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 87 67 58)(2 88 68 59)(3 89 69 60)(4 90 70 46)(5 76 71 47)(6 77 72 48)(7 78 73 49)(8 79 74 50)(9 80 75 51)(10 81 61 52)(11 82 62 53)(12 83 63 54)(13 84 64 55)(14 85 65 56)(15 86 66 57)(16 112 91 44)(17 113 92 45)(18 114 93 31)(19 115 94 32)(20 116 95 33)(21 117 96 34)(22 118 97 35)(23 119 98 36)(24 120 99 37)(25 106 100 38)(26 107 101 39)(27 108 102 40)(28 109 103 41)(29 110 104 42)(30 111 105 43)
(1 22 87 35 67 97 58 118)(2 23 88 36 68 98 59 119)(3 24 89 37 69 99 60 120)(4 25 90 38 70 100 46 106)(5 26 76 39 71 101 47 107)(6 27 77 40 72 102 48 108)(7 28 78 41 73 103 49 109)(8 29 79 42 74 104 50 110)(9 30 80 43 75 105 51 111)(10 16 81 44 61 91 52 112)(11 17 82 45 62 92 53 113)(12 18 83 31 63 93 54 114)(13 19 84 32 64 94 55 115)(14 20 85 33 65 95 56 116)(15 21 86 34 66 96 57 117)
(1 97 87 35 67 22 58 118)(2 98 88 36 68 23 59 119)(3 99 89 37 69 24 60 120)(4 100 90 38 70 25 46 106)(5 101 76 39 71 26 47 107)(6 102 77 40 72 27 48 108)(7 103 78 41 73 28 49 109)(8 104 79 42 74 29 50 110)(9 105 80 43 75 30 51 111)(10 91 81 44 61 16 52 112)(11 92 82 45 62 17 53 113)(12 93 83 31 63 18 54 114)(13 94 84 32 64 19 55 115)(14 95 85 33 65 20 56 116)(15 96 86 34 66 21 57 117)
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), (1,87,67,58)(2,88,68,59)(3,89,69,60)(4,90,70,46)(5,76,71,47)(6,77,72,48)(7,78,73,49)(8,79,74,50)(9,80,75,51)(10,81,61,52)(11,82,62,53)(12,83,63,54)(13,84,64,55)(14,85,65,56)(15,86,66,57)(16,112,91,44)(17,113,92,45)(18,114,93,31)(19,115,94,32)(20,116,95,33)(21,117,96,34)(22,118,97,35)(23,119,98,36)(24,120,99,37)(25,106,100,38)(26,107,101,39)(27,108,102,40)(28,109,103,41)(29,110,104,42)(30,111,105,43), (1,22,87,35,67,97,58,118)(2,23,88,36,68,98,59,119)(3,24,89,37,69,99,60,120)(4,25,90,38,70,100,46,106)(5,26,76,39,71,101,47,107)(6,27,77,40,72,102,48,108)(7,28,78,41,73,103,49,109)(8,29,79,42,74,104,50,110)(9,30,80,43,75,105,51,111)(10,16,81,44,61,91,52,112)(11,17,82,45,62,92,53,113)(12,18,83,31,63,93,54,114)(13,19,84,32,64,94,55,115)(14,20,85,33,65,95,56,116)(15,21,86,34,66,96,57,117), (1,97,87,35,67,22,58,118)(2,98,88,36,68,23,59,119)(3,99,89,37,69,24,60,120)(4,100,90,38,70,25,46,106)(5,101,76,39,71,26,47,107)(6,102,77,40,72,27,48,108)(7,103,78,41,73,28,49,109)(8,104,79,42,74,29,50,110)(9,105,80,43,75,30,51,111)(10,91,81,44,61,16,52,112)(11,92,82,45,62,17,53,113)(12,93,83,31,63,18,54,114)(13,94,84,32,64,19,55,115)(14,95,85,33,65,20,56,116)(15,96,86,34,66,21,57,117)>;
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), (1,87,67,58)(2,88,68,59)(3,89,69,60)(4,90,70,46)(5,76,71,47)(6,77,72,48)(7,78,73,49)(8,79,74,50)(9,80,75,51)(10,81,61,52)(11,82,62,53)(12,83,63,54)(13,84,64,55)(14,85,65,56)(15,86,66,57)(16,112,91,44)(17,113,92,45)(18,114,93,31)(19,115,94,32)(20,116,95,33)(21,117,96,34)(22,118,97,35)(23,119,98,36)(24,120,99,37)(25,106,100,38)(26,107,101,39)(27,108,102,40)(28,109,103,41)(29,110,104,42)(30,111,105,43), (1,22,87,35,67,97,58,118)(2,23,88,36,68,98,59,119)(3,24,89,37,69,99,60,120)(4,25,90,38,70,100,46,106)(5,26,76,39,71,101,47,107)(6,27,77,40,72,102,48,108)(7,28,78,41,73,103,49,109)(8,29,79,42,74,104,50,110)(9,30,80,43,75,105,51,111)(10,16,81,44,61,91,52,112)(11,17,82,45,62,92,53,113)(12,18,83,31,63,93,54,114)(13,19,84,32,64,94,55,115)(14,20,85,33,65,95,56,116)(15,21,86,34,66,96,57,117), (1,97,87,35,67,22,58,118)(2,98,88,36,68,23,59,119)(3,99,89,37,69,24,60,120)(4,100,90,38,70,25,46,106)(5,101,76,39,71,26,47,107)(6,102,77,40,72,27,48,108)(7,103,78,41,73,28,49,109)(8,104,79,42,74,29,50,110)(9,105,80,43,75,30,51,111)(10,91,81,44,61,16,52,112)(11,92,82,45,62,17,53,113)(12,93,83,31,63,18,54,114)(13,94,84,32,64,19,55,115)(14,95,85,33,65,20,56,116)(15,96,86,34,66,21,57,117) );
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)], [(1,87,67,58),(2,88,68,59),(3,89,69,60),(4,90,70,46),(5,76,71,47),(6,77,72,48),(7,78,73,49),(8,79,74,50),(9,80,75,51),(10,81,61,52),(11,82,62,53),(12,83,63,54),(13,84,64,55),(14,85,65,56),(15,86,66,57),(16,112,91,44),(17,113,92,45),(18,114,93,31),(19,115,94,32),(20,116,95,33),(21,117,96,34),(22,118,97,35),(23,119,98,36),(24,120,99,37),(25,106,100,38),(26,107,101,39),(27,108,102,40),(28,109,103,41),(29,110,104,42),(30,111,105,43)], [(1,22,87,35,67,97,58,118),(2,23,88,36,68,98,59,119),(3,24,89,37,69,99,60,120),(4,25,90,38,70,100,46,106),(5,26,76,39,71,101,47,107),(6,27,77,40,72,102,48,108),(7,28,78,41,73,103,49,109),(8,29,79,42,74,104,50,110),(9,30,80,43,75,105,51,111),(10,16,81,44,61,91,52,112),(11,17,82,45,62,92,53,113),(12,18,83,31,63,93,54,114),(13,19,84,32,64,94,55,115),(14,20,85,33,65,95,56,116),(15,21,86,34,66,96,57,117)], [(1,97,87,35,67,22,58,118),(2,98,88,36,68,23,59,119),(3,99,89,37,69,24,60,120),(4,100,90,38,70,25,46,106),(5,101,76,39,71,26,47,107),(6,102,77,40,72,27,48,108),(7,103,78,41,73,28,49,109),(8,104,79,42,74,29,50,110),(9,105,80,43,75,30,51,111),(10,91,81,44,61,16,52,112),(11,92,82,45,62,17,53,113),(12,93,83,31,63,18,54,114),(13,94,84,32,64,19,55,115),(14,95,85,33,65,20,56,116),(15,96,86,34,66,21,57,117)]])
165 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30H | 30I | ··· | 30P | 30Q | ··· | 30AF | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C60 | D4 | C3×D4 | C5×D4 | D4×C15 | C4.D4 | C3×C4.D4 | C5×C4.D4 | C15×C4.D4 |
| kernel | C15×C4.D4 | C15×M4(2) | D4×C30 | C5×C4.D4 | C22×C30 | C3×C4.D4 | C5×M4(2) | D4×C10 | C3×M4(2) | C6×D4 | C22×C10 | C4.D4 | C22×C6 | M4(2) | C2×D4 | C23 | C60 | C20 | C12 | C4 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 4 | 8 | 8 | 16 | 16 | 8 | 32 | 2 | 4 | 8 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C15×C4.D4 ►in GL8(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 87 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 87 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 217 | 129 | 0 | 1 |
| 0 | 0 | 0 | 0 | 129 | 24 | 240 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 106 | 153 | 239 | 0 |
| 0 | 0 | 0 | 0 | 105 | 152 | 0 | 2 |
| 0 | 0 | 0 | 0 | 34 | 95 | 135 | 153 |
| 0 | 0 | 0 | 0 | 71 | 57 | 105 | 89 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 106 | 153 | 239 | 0 |
| 0 | 0 | 0 | 0 | 136 | 89 | 0 | 239 |
| 0 | 0 | 0 | 0 | 116 | 216 | 135 | 88 |
| 0 | 0 | 0 | 0 | 126 | 25 | 105 | 152 |
G:=sub<GL(8,GF(241))| [15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,87,0,0,0,0,0,0,0,0,87,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,217,129,0,0,0,0,1,0,129,24,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,0,106,105,34,71,0,0,0,0,153,152,95,57,0,0,0,0,239,0,135,105,0,0,0,0,0,2,153,89],[0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,106,136,116,126,0,0,0,0,153,89,216,25,0,0,0,0,239,0,135,105,0,0,0,0,0,239,88,152] >;
C15×C4.D4 in GAP, Magma, Sage, TeX
C_{15}\times C_4.D_4 % in TeX
G:=Group("C15xC4.D4"); // GroupNames label
G:=SmallGroup(480,203);
// by ID
G=gap.SmallGroup(480,203);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,10504,7572,124]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations