direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C8⋊D5, C40⋊4C6, C24⋊7D5, C120⋊10C2, D10.1C12, C12.57D10, C15⋊11M4(2), C60.70C22, Dic5.1C12, C8⋊3(C3×D5), C5⋊2C8⋊4C6, (C6×D5).3C4, (C4×D5).2C6, C2.3(D5×C12), C4.13(C6×D5), C6.16(C4×D5), C5⋊3(C3×M4(2)), C30.41(C2×C4), C10.9(C2×C12), C20.14(C2×C6), (D5×C12).7C2, (C3×Dic5).3C4, (C3×C5⋊2C8)⋊11C2, SmallGroup(240,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊D5
G = < a,b,c,d | a3=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Smallest permutation representation of C3×C8⋊D5
►On 120 points
Generators in S120
(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 56)(10 87 49)(11 88 50)(12 81 51)(13 82 52)(14 83 53)(15 84 54)(16 85 55)(17 90 115)(18 91 116)(19 92 117)(20 93 118)(21 94 119)(22 95 120)(23 96 113)(24 89 114)(25 48 66)(26 41 67)(27 42 68)(28 43 69)(29 44 70)(30 45 71)(31 46 72)(32 47 65)(33 108 74)(34 109 75)(35 110 76)(36 111 77)(37 112 78)(38 105 79)(39 106 80)(40 107 73)
(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 50 19 46 109)(2 51 20 47 110)(3 52 21 48 111)(4 53 22 41 112)(5 54 23 42 105)(6 55 24 43 106)(7 56 17 44 107)(8 49 18 45 108)(9 90 70 73 98)(10 91 71 74 99)(11 92 72 75 100)(12 93 65 76 101)(13 94 66 77 102)(14 95 67 78 103)(15 96 68 79 104)(16 89 69 80 97)(25 36 62 82 119)(26 37 63 83 120)(27 38 64 84 113)(28 39 57 85 114)(29 40 58 86 115)(30 33 59 87 116)(31 34 60 88 117)(32 35 61 81 118)
(1 109)(2 106)(3 111)(4 108)(5 105)(6 110)(7 107)(8 112)(9 70)(10 67)(11 72)(12 69)(13 66)(14 71)(15 68)(16 65)(18 22)(20 24)(25 82)(26 87)(27 84)(28 81)(29 86)(30 83)(31 88)(32 85)(33 63)(34 60)(35 57)(36 62)(37 59)(38 64)(39 61)(40 58)(41 49)(42 54)(43 51)(44 56)(45 53)(46 50)(47 55)(48 52)(73 98)(74 103)(75 100)(76 97)(77 102)(78 99)(79 104)(80 101)(89 93)(91 95)(114 118)(116 120)
G:=sub<Sym(120)| (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,56)(10,87,49)(11,88,50)(12,81,51)(13,82,52)(14,83,53)(15,84,54)(16,85,55)(17,90,115)(18,91,116)(19,92,117)(20,93,118)(21,94,119)(22,95,120)(23,96,113)(24,89,114)(25,48,66)(26,41,67)(27,42,68)(28,43,69)(29,44,70)(30,45,71)(31,46,72)(32,47,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (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), (1,50,19,46,109)(2,51,20,47,110)(3,52,21,48,111)(4,53,22,41,112)(5,54,23,42,105)(6,55,24,43,106)(7,56,17,44,107)(8,49,18,45,108)(9,90,70,73,98)(10,91,71,74,99)(11,92,72,75,100)(12,93,65,76,101)(13,94,66,77,102)(14,95,67,78,103)(15,96,68,79,104)(16,89,69,80,97)(25,36,62,82,119)(26,37,63,83,120)(27,38,64,84,113)(28,39,57,85,114)(29,40,58,86,115)(30,33,59,87,116)(31,34,60,88,117)(32,35,61,81,118), (1,109)(2,106)(3,111)(4,108)(5,105)(6,110)(7,107)(8,112)(9,70)(10,67)(11,72)(12,69)(13,66)(14,71)(15,68)(16,65)(18,22)(20,24)(25,82)(26,87)(27,84)(28,81)(29,86)(30,83)(31,88)(32,85)(33,63)(34,60)(35,57)(36,62)(37,59)(38,64)(39,61)(40,58)(41,49)(42,54)(43,51)(44,56)(45,53)(46,50)(47,55)(48,52)(73,98)(74,103)(75,100)(76,97)(77,102)(78,99)(79,104)(80,101)(89,93)(91,95)(114,118)(116,120)>;
G:=Group( (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,56)(10,87,49)(11,88,50)(12,81,51)(13,82,52)(14,83,53)(15,84,54)(16,85,55)(17,90,115)(18,91,116)(19,92,117)(20,93,118)(21,94,119)(22,95,120)(23,96,113)(24,89,114)(25,48,66)(26,41,67)(27,42,68)(28,43,69)(29,44,70)(30,45,71)(31,46,72)(32,47,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (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), (1,50,19,46,109)(2,51,20,47,110)(3,52,21,48,111)(4,53,22,41,112)(5,54,23,42,105)(6,55,24,43,106)(7,56,17,44,107)(8,49,18,45,108)(9,90,70,73,98)(10,91,71,74,99)(11,92,72,75,100)(12,93,65,76,101)(13,94,66,77,102)(14,95,67,78,103)(15,96,68,79,104)(16,89,69,80,97)(25,36,62,82,119)(26,37,63,83,120)(27,38,64,84,113)(28,39,57,85,114)(29,40,58,86,115)(30,33,59,87,116)(31,34,60,88,117)(32,35,61,81,118), (1,109)(2,106)(3,111)(4,108)(5,105)(6,110)(7,107)(8,112)(9,70)(10,67)(11,72)(12,69)(13,66)(14,71)(15,68)(16,65)(18,22)(20,24)(25,82)(26,87)(27,84)(28,81)(29,86)(30,83)(31,88)(32,85)(33,63)(34,60)(35,57)(36,62)(37,59)(38,64)(39,61)(40,58)(41,49)(42,54)(43,51)(44,56)(45,53)(46,50)(47,55)(48,52)(73,98)(74,103)(75,100)(76,97)(77,102)(78,99)(79,104)(80,101)(89,93)(91,95)(114,118)(116,120) );
G=PermutationGroup([[(1,100,60),(2,101,61),(3,102,62),(4,103,63),(5,104,64),(6,97,57),(7,98,58),(8,99,59),(9,86,56),(10,87,49),(11,88,50),(12,81,51),(13,82,52),(14,83,53),(15,84,54),(16,85,55),(17,90,115),(18,91,116),(19,92,117),(20,93,118),(21,94,119),(22,95,120),(23,96,113),(24,89,114),(25,48,66),(26,41,67),(27,42,68),(28,43,69),(29,44,70),(30,45,71),(31,46,72),(32,47,65),(33,108,74),(34,109,75),(35,110,76),(36,111,77),(37,112,78),(38,105,79),(39,106,80),(40,107,73)], [(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)], [(1,50,19,46,109),(2,51,20,47,110),(3,52,21,48,111),(4,53,22,41,112),(5,54,23,42,105),(6,55,24,43,106),(7,56,17,44,107),(8,49,18,45,108),(9,90,70,73,98),(10,91,71,74,99),(11,92,72,75,100),(12,93,65,76,101),(13,94,66,77,102),(14,95,67,78,103),(15,96,68,79,104),(16,89,69,80,97),(25,36,62,82,119),(26,37,63,83,120),(27,38,64,84,113),(28,39,57,85,114),(29,40,58,86,115),(30,33,59,87,116),(31,34,60,88,117),(32,35,61,81,118)], [(1,109),(2,106),(3,111),(4,108),(5,105),(6,110),(7,107),(8,112),(9,70),(10,67),(11,72),(12,69),(13,66),(14,71),(15,68),(16,65),(18,22),(20,24),(25,82),(26,87),(27,84),(28,81),(29,86),(30,83),(31,88),(32,85),(33,63),(34,60),(35,57),(36,62),(37,59),(38,64),(39,61),(40,58),(41,49),(42,54),(43,51),(44,56),(45,53),(46,50),(47,55),(48,52),(73,98),(74,103),(75,100),(76,97),(77,102),(78,99),(79,104),(80,101),(89,93),(91,95),(114,118),(116,120)]])
C3×C8⋊D5 is a maximal subgroup of
C40⋊D6 C24⋊D10 D24⋊D5 Dic60⋊C2 C24.2D10 C40.55D6 C40.35D6 C3×D5×M4(2)
78 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 10A | 10B | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 10 | 1 | 1 | 1 | 1 | 10 | 2 | 2 | 1 | 1 | 10 | 10 | 2 | 2 | 10 | 10 | 2 | 2 | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D5 | M4(2) | D10 | C3×D5 | C4×D5 | C3×M4(2) | C6×D5 | C8⋊D5 | D5×C12 | C3×C8⋊D5 |
| kernel | C3×C8⋊D5 | C3×C5⋊2C8 | C120 | D5×C12 | C8⋊D5 | C3×Dic5 | C6×D5 | C5⋊2C8 | C40 | C4×D5 | Dic5 | D10 | C24 | C15 | C12 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×C8⋊D5 ►in GL3(𝔽241) generated by
| 225 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 216 | 140 |
| 0 | 101 | 25 |
| 1 | 0 | 0 |
| 0 | 0 | 240 |
| 0 | 1 | 189 |
| 240 | 0 | 0 |
| 0 | 52 | 240 |
| 0 | 52 | 189 |
G:=sub<GL(3,GF(241))| [225,0,0,0,1,0,0,0,1],[1,0,0,0,216,101,0,140,25],[1,0,0,0,0,1,0,240,189],[240,0,0,0,52,52,0,240,189] >;
C3×C8⋊D5 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes D_5
% in TeX
G:=Group("C3xC8:D5"); // GroupNames label
G:=SmallGroup(240,34);
// by ID
G=gap.SmallGroup(240,34);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-5,313,79,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations
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