direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C3⋊C8, C20.29D6, C12.29D10, C60.29C22, D10.4Dic3, Dic5.4Dic3, C3⋊3(C8×D5), C15⋊5(C2×C8), (C3×D5)⋊1C8, C15⋊3C8⋊9C2, (C6×D5).1C4, (C4×D5).7S3, C6.10(C4×D5), C4.22(S3×D5), C30.22(C2×C4), (D5×C12).2C2, C2.1(D5×Dic3), (C3×Dic5).1C4, C10.8(C2×Dic3), C5⋊3(C2×C3⋊C8), (C5×C3⋊C8)⋊4C2, SmallGroup(240,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — D5×C3⋊C8 |
Generators and relations for D5×C3⋊C8
G = < a,b,c,d | a5=b2=c3=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of D5×C3⋊C8
►On 120 points
Generators in S120
(1 28 113 88 99)(2 29 114 81 100)(3 30 115 82 101)(4 31 116 83 102)(5 32 117 84 103)(6 25 118 85 104)(7 26 119 86 97)(8 27 120 87 98)(9 22 39 72 43)(10 23 40 65 44)(11 24 33 66 45)(12 17 34 67 46)(13 18 35 68 47)(14 19 36 69 48)(15 20 37 70 41)(16 21 38 71 42)(49 112 78 95 61)(50 105 79 96 62)(51 106 80 89 63)(52 107 73 90 64)(53 108 74 91 57)(54 109 75 92 58)(55 110 76 93 59)(56 111 77 94 60)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 65)(24 66)(25 85)(26 86)(27 87)(28 88)(29 81)(30 82)(31 83)(32 84)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 105)
(1 11 50)(2 51 12)(3 13 52)(4 53 14)(5 15 54)(6 55 16)(7 9 56)(8 49 10)(17 29 106)(18 107 30)(19 31 108)(20 109 32)(21 25 110)(22 111 26)(23 27 112)(24 105 28)(33 79 113)(34 114 80)(35 73 115)(36 116 74)(37 75 117)(38 118 76)(39 77 119)(40 120 78)(41 58 103)(42 104 59)(43 60 97)(44 98 61)(45 62 99)(46 100 63)(47 64 101)(48 102 57)(65 87 95)(66 96 88)(67 81 89)(68 90 82)(69 83 91)(70 92 84)(71 85 93)(72 94 86)
(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)
G:=sub<Sym(120)| (1,28,113,88,99)(2,29,114,81,100)(3,30,115,82,101)(4,31,116,83,102)(5,32,117,84,103)(6,25,118,85,104)(7,26,119,86,97)(8,27,120,87,98)(9,22,39,72,43)(10,23,40,65,44)(11,24,33,66,45)(12,17,34,67,46)(13,18,35,68,47)(14,19,36,69,48)(15,20,37,70,41)(16,21,38,71,42)(49,112,78,95,61)(50,105,79,96,62)(51,106,80,89,63)(52,107,73,90,64)(53,108,74,91,57)(54,109,75,92,58)(55,110,76,93,59)(56,111,77,94,60), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,65)(24,66)(25,85)(26,86)(27,87)(28,88)(29,81)(30,82)(31,83)(32,84)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,105), (1,11,50)(2,51,12)(3,13,52)(4,53,14)(5,15,54)(6,55,16)(7,9,56)(8,49,10)(17,29,106)(18,107,30)(19,31,108)(20,109,32)(21,25,110)(22,111,26)(23,27,112)(24,105,28)(33,79,113)(34,114,80)(35,73,115)(36,116,74)(37,75,117)(38,118,76)(39,77,119)(40,120,78)(41,58,103)(42,104,59)(43,60,97)(44,98,61)(45,62,99)(46,100,63)(47,64,101)(48,102,57)(65,87,95)(66,96,88)(67,81,89)(68,90,82)(69,83,91)(70,92,84)(71,85,93)(72,94,86), (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)>;
G:=Group( (1,28,113,88,99)(2,29,114,81,100)(3,30,115,82,101)(4,31,116,83,102)(5,32,117,84,103)(6,25,118,85,104)(7,26,119,86,97)(8,27,120,87,98)(9,22,39,72,43)(10,23,40,65,44)(11,24,33,66,45)(12,17,34,67,46)(13,18,35,68,47)(14,19,36,69,48)(15,20,37,70,41)(16,21,38,71,42)(49,112,78,95,61)(50,105,79,96,62)(51,106,80,89,63)(52,107,73,90,64)(53,108,74,91,57)(54,109,75,92,58)(55,110,76,93,59)(56,111,77,94,60), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,67)(18,68)(19,69)(20,70)(21,71)(22,72)(23,65)(24,66)(25,85)(26,86)(27,87)(28,88)(29,81)(30,82)(31,83)(32,84)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,105), (1,11,50)(2,51,12)(3,13,52)(4,53,14)(5,15,54)(6,55,16)(7,9,56)(8,49,10)(17,29,106)(18,107,30)(19,31,108)(20,109,32)(21,25,110)(22,111,26)(23,27,112)(24,105,28)(33,79,113)(34,114,80)(35,73,115)(36,116,74)(37,75,117)(38,118,76)(39,77,119)(40,120,78)(41,58,103)(42,104,59)(43,60,97)(44,98,61)(45,62,99)(46,100,63)(47,64,101)(48,102,57)(65,87,95)(66,96,88)(67,81,89)(68,90,82)(69,83,91)(70,92,84)(71,85,93)(72,94,86), (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) );
G=PermutationGroup([[(1,28,113,88,99),(2,29,114,81,100),(3,30,115,82,101),(4,31,116,83,102),(5,32,117,84,103),(6,25,118,85,104),(7,26,119,86,97),(8,27,120,87,98),(9,22,39,72,43),(10,23,40,65,44),(11,24,33,66,45),(12,17,34,67,46),(13,18,35,68,47),(14,19,36,69,48),(15,20,37,70,41),(16,21,38,71,42),(49,112,78,95,61),(50,105,79,96,62),(51,106,80,89,63),(52,107,73,90,64),(53,108,74,91,57),(54,109,75,92,58),(55,110,76,93,59),(56,111,77,94,60)], [(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,97),(8,98),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,41),(16,42),(17,67),(18,68),(19,69),(20,70),(21,71),(22,72),(23,65),(24,66),(25,85),(26,86),(27,87),(28,88),(29,81),(30,82),(31,83),(32,84),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60),(89,106),(90,107),(91,108),(92,109),(93,110),(94,111),(95,112),(96,105)], [(1,11,50),(2,51,12),(3,13,52),(4,53,14),(5,15,54),(6,55,16),(7,9,56),(8,49,10),(17,29,106),(18,107,30),(19,31,108),(20,109,32),(21,25,110),(22,111,26),(23,27,112),(24,105,28),(33,79,113),(34,114,80),(35,73,115),(36,116,74),(37,75,117),(38,118,76),(39,77,119),(40,120,78),(41,58,103),(42,104,59),(43,60,97),(44,98,61),(45,62,99),(46,100,63),(47,64,101),(48,102,57),(65,87,95),(66,96,88),(67,81,89),(68,90,82),(69,83,91),(70,92,84),(71,85,93),(72,94,86)], [(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)]])
D5×C3⋊C8 is a maximal subgroup of
C30.C42 C30.3C42 C30.4C42 Dic5.Dic6 Dic5.4Dic6 D10.Dic6 D10.2Dic6 S3×C8×D5 C40.55D6 C40.35D6 D20.3Dic3 D20.2Dic3 D12.24D10 C60.16C23 D20.14D6 D20.D6
D5×C3⋊C8 is a maximal quotient of
C40.51D6 C60.93D4 C60.13Q8
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 10 | 10 | 3 | 3 | 3 | 3 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D5 | Dic3 | D6 | Dic3 | D10 | C3⋊C8 | C4×D5 | C8×D5 | S3×D5 | D5×Dic3 | D5×C3⋊C8 |
| kernel | D5×C3⋊C8 | C5×C3⋊C8 | C15⋊3C8 | D5×C12 | C3×Dic5 | C6×D5 | C3×D5 | C4×D5 | C3⋊C8 | Dic5 | C20 | D10 | C12 | D5 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 8 | 2 | 2 | 4 |
Matrix representation of D5×C3⋊C8 ►in GL4(𝔽241) generated by
| 240 | 1 | 0 | 0 |
| 188 | 52 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 188 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 240 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 166 | 126 |
| 0 | 0 | 51 | 75 |
G:=sub<GL(4,GF(241))| [240,188,0,0,1,52,0,0,0,0,1,0,0,0,0,1],[240,188,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,240,240],[30,0,0,0,0,30,0,0,0,0,166,51,0,0,126,75] >;
D5×C3⋊C8 in GAP, Magma, Sage, TeX
D_5\times C_3\rtimes C_8
% in TeX
G:=Group("D5xC3:C8"); // GroupNames label
G:=SmallGroup(240,7);
// by ID
G=gap.SmallGroup(240,7);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,31,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^3=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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