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## G = C15×C4.D4order 480 = 25·3·5

### Direct product of C15 and C4.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C15×C4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C2×C60 — C15×M4(2) — C15×C4.D4
 Lower central C1 — C2 — C22 — C15×C4.D4
 Upper central C1 — C30 — C2×C60 — C15×C4.D4

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

Smallest permutation representation of 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

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