Copied to
clipboard

## 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 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], C2×C6, C2×C6 [×4], C15, M4(2) [×2], C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C24 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C30, C30 [×3], C4.D4, C40 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C3×M4(2) [×2], C6×D4, C60 [×2], C2×C30, C2×C30 [×4], C5×M4(2) [×2], D4×C10, C3×C4.D4, C120 [×2], C2×C60, D4×C15 [×2], C22×C30 [×2], C5×C4.D4, C15×M4(2) [×2], D4×C30, C15×C4.D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C2×C4, D4 [×2], C10 [×3], C12 [×2], C2×C6, C15, C22⋊C4, C20 [×2], C2×C10, C2×C12, C3×D4 [×2], C30 [×3], C4.D4, C2×C20, C5×D4 [×2], C3×C22⋊C4, C60 [×2], C2×C30, C5×C22⋊C4, C3×C4.D4, C2×C60, D4×C15 [×2], 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 28 120 46)(2 29 106 47)(3 30 107 48)(4 16 108 49)(5 17 109 50)(6 18 110 51)(7 19 111 52)(8 20 112 53)(9 21 113 54)(10 22 114 55)(11 23 115 56)(12 24 116 57)(13 25 117 58)(14 26 118 59)(15 27 119 60)(31 68 82 97)(32 69 83 98)(33 70 84 99)(34 71 85 100)(35 72 86 101)(36 73 87 102)(37 74 88 103)(38 75 89 104)(39 61 90 105)(40 62 76 91)(41 63 77 92)(42 64 78 93)(43 65 79 94)(44 66 80 95)(45 67 81 96)
(1 45 28 96 120 81 46 67)(2 31 29 97 106 82 47 68)(3 32 30 98 107 83 48 69)(4 33 16 99 108 84 49 70)(5 34 17 100 109 85 50 71)(6 35 18 101 110 86 51 72)(7 36 19 102 111 87 52 73)(8 37 20 103 112 88 53 74)(9 38 21 104 113 89 54 75)(10 39 22 105 114 90 55 61)(11 40 23 91 115 76 56 62)(12 41 24 92 116 77 57 63)(13 42 25 93 117 78 58 64)(14 43 26 94 118 79 59 65)(15 44 27 95 119 80 60 66)
(1 81 28 96 120 45 46 67)(2 82 29 97 106 31 47 68)(3 83 30 98 107 32 48 69)(4 84 16 99 108 33 49 70)(5 85 17 100 109 34 50 71)(6 86 18 101 110 35 51 72)(7 87 19 102 111 36 52 73)(8 88 20 103 112 37 53 74)(9 89 21 104 113 38 54 75)(10 90 22 105 114 39 55 61)(11 76 23 91 115 40 56 62)(12 77 24 92 116 41 57 63)(13 78 25 93 117 42 58 64)(14 79 26 94 118 43 59 65)(15 80 27 95 119 44 60 66)

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), (1,28,120,46)(2,29,106,47)(3,30,107,48)(4,16,108,49)(5,17,109,50)(6,18,110,51)(7,19,111,52)(8,20,112,53)(9,21,113,54)(10,22,114,55)(11,23,115,56)(12,24,116,57)(13,25,117,58)(14,26,118,59)(15,27,119,60)(31,68,82,97)(32,69,83,98)(33,70,84,99)(34,71,85,100)(35,72,86,101)(36,73,87,102)(37,74,88,103)(38,75,89,104)(39,61,90,105)(40,62,76,91)(41,63,77,92)(42,64,78,93)(43,65,79,94)(44,66,80,95)(45,67,81,96), (1,45,28,96,120,81,46,67)(2,31,29,97,106,82,47,68)(3,32,30,98,107,83,48,69)(4,33,16,99,108,84,49,70)(5,34,17,100,109,85,50,71)(6,35,18,101,110,86,51,72)(7,36,19,102,111,87,52,73)(8,37,20,103,112,88,53,74)(9,38,21,104,113,89,54,75)(10,39,22,105,114,90,55,61)(11,40,23,91,115,76,56,62)(12,41,24,92,116,77,57,63)(13,42,25,93,117,78,58,64)(14,43,26,94,118,79,59,65)(15,44,27,95,119,80,60,66), (1,81,28,96,120,45,46,67)(2,82,29,97,106,31,47,68)(3,83,30,98,107,32,48,69)(4,84,16,99,108,33,49,70)(5,85,17,100,109,34,50,71)(6,86,18,101,110,35,51,72)(7,87,19,102,111,36,52,73)(8,88,20,103,112,37,53,74)(9,89,21,104,113,38,54,75)(10,90,22,105,114,39,55,61)(11,76,23,91,115,40,56,62)(12,77,24,92,116,41,57,63)(13,78,25,93,117,42,58,64)(14,79,26,94,118,43,59,65)(15,80,27,95,119,44,60,66)>;

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), (1,28,120,46)(2,29,106,47)(3,30,107,48)(4,16,108,49)(5,17,109,50)(6,18,110,51)(7,19,111,52)(8,20,112,53)(9,21,113,54)(10,22,114,55)(11,23,115,56)(12,24,116,57)(13,25,117,58)(14,26,118,59)(15,27,119,60)(31,68,82,97)(32,69,83,98)(33,70,84,99)(34,71,85,100)(35,72,86,101)(36,73,87,102)(37,74,88,103)(38,75,89,104)(39,61,90,105)(40,62,76,91)(41,63,77,92)(42,64,78,93)(43,65,79,94)(44,66,80,95)(45,67,81,96), (1,45,28,96,120,81,46,67)(2,31,29,97,106,82,47,68)(3,32,30,98,107,83,48,69)(4,33,16,99,108,84,49,70)(5,34,17,100,109,85,50,71)(6,35,18,101,110,86,51,72)(7,36,19,102,111,87,52,73)(8,37,20,103,112,88,53,74)(9,38,21,104,113,89,54,75)(10,39,22,105,114,90,55,61)(11,40,23,91,115,76,56,62)(12,41,24,92,116,77,57,63)(13,42,25,93,117,78,58,64)(14,43,26,94,118,79,59,65)(15,44,27,95,119,80,60,66), (1,81,28,96,120,45,46,67)(2,82,29,97,106,31,47,68)(3,83,30,98,107,32,48,69)(4,84,16,99,108,33,49,70)(5,85,17,100,109,34,50,71)(6,86,18,101,110,35,51,72)(7,87,19,102,111,36,52,73)(8,88,20,103,112,37,53,74)(9,89,21,104,113,38,54,75)(10,90,22,105,114,39,55,61)(11,76,23,91,115,40,56,62)(12,77,24,92,116,41,57,63)(13,78,25,93,117,42,58,64)(14,79,26,94,118,43,59,65)(15,80,27,95,119,44,60,66) );

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)], [(1,28,120,46),(2,29,106,47),(3,30,107,48),(4,16,108,49),(5,17,109,50),(6,18,110,51),(7,19,111,52),(8,20,112,53),(9,21,113,54),(10,22,114,55),(11,23,115,56),(12,24,116,57),(13,25,117,58),(14,26,118,59),(15,27,119,60),(31,68,82,97),(32,69,83,98),(33,70,84,99),(34,71,85,100),(35,72,86,101),(36,73,87,102),(37,74,88,103),(38,75,89,104),(39,61,90,105),(40,62,76,91),(41,63,77,92),(42,64,78,93),(43,65,79,94),(44,66,80,95),(45,67,81,96)], [(1,45,28,96,120,81,46,67),(2,31,29,97,106,82,47,68),(3,32,30,98,107,83,48,69),(4,33,16,99,108,84,49,70),(5,34,17,100,109,85,50,71),(6,35,18,101,110,86,51,72),(7,36,19,102,111,87,52,73),(8,37,20,103,112,88,53,74),(9,38,21,104,113,89,54,75),(10,39,22,105,114,90,55,61),(11,40,23,91,115,76,56,62),(12,41,24,92,116,77,57,63),(13,42,25,93,117,78,58,64),(14,43,26,94,118,79,59,65),(15,44,27,95,119,80,60,66)], [(1,81,28,96,120,45,46,67),(2,82,29,97,106,31,47,68),(3,83,30,98,107,32,48,69),(4,84,16,99,108,33,49,70),(5,85,17,100,109,34,50,71),(6,86,18,101,110,35,51,72),(7,87,19,102,111,36,52,73),(8,88,20,103,112,37,53,74),(9,89,21,104,113,38,54,75),(10,90,22,105,114,39,55,61),(11,76,23,91,115,40,56,62),(12,77,24,92,116,41,57,63),(13,78,25,93,117,42,58,64),(14,79,26,94,118,43,59,65),(15,80,27,95,119,44,60,66)])

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

׿
×
𝔽