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G = C8×D15order 240 = 24·3·5

Direct product of C8 and D15

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C15 — C8×D15
C1C5C15C30C60C4×D15 — C8×D15
C15 — C8×D15
C1C8

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

15C2
15C2
15C22
15C4
5S3
5S3
3D5
3D5
15C8
15C2×C4
5Dic3
5D6
3D10
3Dic5
15C2×C8
5C3⋊C8
5C4×S3
3C4×D5
3C52C8
5S3×C8
3C8×D5

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

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

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

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

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 2A2B2C 3 4A4B4C4D5A5B 6 8A8B8C8D8E8F8G8H10A10B12A12B15A15B15C15D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H60A···60H120A···120P
order12223444455688888888101012121515151520202020242424243030303040···4060···60120···120
size1115152111515222111115151515222222222222222222222···22···22···2

72 irreducible representations

dim1111111222222222222
type++++++++++
imageC1C2C2C2C4C4C8S3D5D6D10C4×S3D15C4×D5S3×C8D30C8×D5C4×D15C8×D15
kernelC8×D15C153C8C120C4×D15Dic15D30D15C40C24C20C12C10C8C6C5C4C3C2C1
# reps11112281212244448816

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

211000
021100
0010
0001
,
2405100
19019000
001192
0064239
,
2405100
0100
001192
000240
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

Export

Subgroup lattice of C8×D15 in TeX

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