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## G = C8×D15order 240 = 24·3·5

### Direct product of C8 and D15

Aliases: C8×D15, C403S3, C244D5, C1204C2, D30.6C4, C20.47D6, C4.12D30, C12.48D10, C60.54C22, Dic15.6C4, C54(S3×C8), C32(C8×D5), C1510(C2×C8), C6.5(C4×D5), C2.1(C4×D15), C153C813C2, C10.12(C4×S3), C30.35(C2×C4), (C4×D15).6C2, SmallGroup(240,65)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C8×D15
 Chief series C1 — C5 — C15 — C30 — C60 — C4×D15 — C8×D15
 Lower central C15 — C8×D15
 Upper central C1 — C8

Generators and relations for C8×D15
G = < a,b,c | a8=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C8×D15
On 120 points
Generators in S120
(1 107 50 88 28 103 43 61)(2 108 51 89 29 104 44 62)(3 109 52 90 30 105 45 63)(4 110 53 76 16 91 31 64)(5 111 54 77 17 92 32 65)(6 112 55 78 18 93 33 66)(7 113 56 79 19 94 34 67)(8 114 57 80 20 95 35 68)(9 115 58 81 21 96 36 69)(10 116 59 82 22 97 37 70)(11 117 60 83 23 98 38 71)(12 118 46 84 24 99 39 72)(13 119 47 85 25 100 40 73)(14 120 48 86 26 101 41 74)(15 106 49 87 27 102 42 75)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)(25 30)(26 29)(27 28)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 53)(47 52)(48 51)(49 50)(54 60)(55 59)(56 58)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 84)(77 83)(78 82)(79 81)(85 90)(86 89)(87 88)(91 99)(92 98)(93 97)(94 96)(100 105)(101 104)(102 103)(106 107)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)

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

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

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

C8×D15 is a maximal subgroup of
D152C16  D30.5C8  C80⋊S3  S3×C8×D5  C40⋊D6  C4014D6  C405D6  Dic10.D6  C40.54D6  C40.35D6  Dic6.D10  D405S3  D245D5  D60.6C4  D60.3C4  D83D15  D4.5D30  D1208C2
C8×D15 is a maximal quotient of
C80⋊S3  C60.26Q8  D303C8

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40H 60A ··· 60H 120A ··· 120P order 1 2 2 2 3 4 4 4 4 5 5 6 8 8 8 8 8 8 8 8 10 10 12 12 15 15 15 15 20 20 20 20 24 24 24 24 30 30 30 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 15 15 2 1 1 15 15 2 2 2 1 1 1 1 15 15 15 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 D5 D6 D10 C4×S3 D15 C4×D5 S3×C8 D30 C8×D5 C4×D15 C8×D15 kernel C8×D15 C15⋊3C8 C120 C4×D15 Dic15 D30 D15 C40 C24 C20 C12 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 2 2 8 1 2 1 2 2 4 4 4 4 8 8 16

Matrix representation of C8×D15 in GL4(𝔽241) generated by

 211 0 0 0 0 211 0 0 0 0 1 0 0 0 0 1
,
 240 51 0 0 190 190 0 0 0 0 1 192 0 0 64 239
,
 240 51 0 0 0 1 0 0 0 0 1 192 0 0 0 240
G:=sub<GL(4,GF(241))| [211,0,0,0,0,211,0,0,0,0,1,0,0,0,0,1],[240,190,0,0,51,190,0,0,0,0,1,64,0,0,192,239],[240,0,0,0,51,1,0,0,0,0,1,0,0,0,192,240] >;

C8×D15 in GAP, Magma, Sage, TeX

C_8\times D_{15}
% in TeX

G:=Group("C8xD15");
// GroupNames label

G:=SmallGroup(240,65);
// by ID

G=gap.SmallGroup(240,65);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,31,50,964,6917]);
// Polycyclic

G:=Group<a,b,c|a^8=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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