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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

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

C1C22 — C15×C4.D4
C1C2C4C2×C4C2×C20C2×C60C15×M4(2) — C15×C4.D4
C1C2C22 — C15×C4.D4
C1C30C2×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 2A2B2C2D3A3B4A4B5A5B5C5D6A6B6C6D6E6F6G6H8A8B8C8D10A10B10C10D10E10F10G10H10I···10P12A12B12C12D15A···15H20A···20H24A···24H30A···30H30I···30P30Q···30AF40A···40P60A···60P120A···120AF
order1222233445555666666668888101010101010101010···101212121215···1520···2024···2430···3030···3030···3040···4060···60120···120
size1124411221111112244444444111122224···422221···12···24···41···12···24···44···42···24···4

165 irreducible representations

dim111111111111111122224444
type+++++
imageC1C2C2C3C4C5C6C6C10C10C12C15C20C30C30C60D4C3×D4C5×D4D4×C15C4.D4C3×C4.D4C5×C4.D4C15×C4.D4
kernelC15×C4.D4C15×M4(2)D4×C30C5×C4.D4C22×C30C3×C4.D4C5×M4(2)D4×C10C3×M4(2)C6×D4C22×C10C4.D4C22×C6M4(2)C2×D4C23C60C20C12C4C15C5C3C1
# reps1212444284881616832248161248

Matrix representation of C15×C4.D4 in GL8(𝔽241)

150000000
015000000
008700000
000870000
00001000
00000100
00000010
00000001
,
2400000000
0240000000
00100000
00010000
00000100
0000240000
000021712901
0000129242400
,
01000000
10000000
0001770000
0017700000
00001061532390
000010515202
00003495135153
0000715710589
,
0240000000
10000000
000640000
0017700000
00001061532390
0000136890239
000011621613588
000012625105152

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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