direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D15, C40⋊3S3, C24⋊4D5, C120⋊4C2, D30.6C4, C20.47D6, C4.12D30, C12.48D10, C60.54C22, Dic15.6C4, C5⋊4(S3×C8), C3⋊2(C8×D5), C15⋊10(C2×C8), C6.5(C4×D5), C2.1(C4×D15), C15⋊3C8⋊13C2, C10.12(C4×S3), C30.35(C2×C4), (C4×D15).6C2, SmallGroup(240,65)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C8×D15 |
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
D15⋊2C16 D30.5C8 C80⋊S3 S3×C8×D5 C40⋊D6 C40⋊14D6 C40⋊5D6 Dic10.D6 C40.54D6 C40.35D6 Dic6.D10 D40⋊5S3 D24⋊5D5 D60.6C4 D60.3C4 D8⋊3D15 D4.5D30 D120⋊8C2
C8×D15 is a maximal quotient of
C80⋊S3 C60.26Q8 D30⋊3C8
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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