direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C8⋊D6, C40⋊19D6, D24⋊2C10, C20.66D12, C60.145D4, C120⋊26C22, C60.271C23, C8⋊1(S3×C10), C24⋊1(C2×C10), C24⋊C2⋊1C10, C4○D12⋊2C10, (C5×D24)⋊10C2, (C2×D12)⋊7C10, D12⋊4(C2×C10), (C2×C30).92D4, C4.14(C5×D12), C12.12(C5×D4), C6.13(D4×C10), (C10×D12)⋊23C2, C15⋊27(C8⋊C22), Dic6⋊4(C2×C10), C2.15(C10×D12), (C2×C20).240D6, C30.300(C2×D4), C10.84(C2×D12), (C2×C10).27D12, (C5×M4(2))⋊5S3, M4(2)⋊1(C5×S3), C22.5(C5×D12), (C5×D12)⋊34C22, (C3×M4(2))⋊1C10, (C15×M4(2))⋊7C2, (C2×C60).355C22, C20.235(C22×S3), C12.32(C22×C10), (C5×Dic6)⋊31C22, C3⋊1(C5×C8⋊C22), C4.32(S3×C2×C10), (C2×C6).5(C5×D4), (C5×C24⋊C2)⋊9C2, (C5×C4○D12)⋊12C2, (C2×C4).13(S3×C10), (C2×C12).28(C2×C10), SmallGroup(480,787)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊D6
G = < a,b,c,d | a5=b8=c6=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >
Subgroups: 420 in 136 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, Dic3, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C2×C10, C24, Dic6, C4×S3, D12, D12, D12, C3⋊D4, C2×C12, C22×S3, C5×S3, C30, C30, C8⋊C22, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C24⋊C2, D24, C3×M4(2), C2×D12, C4○D12, C5×Dic3, C60, S3×C10, C2×C30, C5×M4(2), C5×D8, C5×SD16, D4×C10, C5×C4○D4, C8⋊D6, C120, C5×Dic6, S3×C20, C5×D12, C5×D12, C5×D12, C5×C3⋊D4, C2×C60, S3×C2×C10, C5×C8⋊C22, C5×C24⋊C2, C5×D24, C15×M4(2), C10×D12, C5×C4○D12, C5×C8⋊D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, D12, C22×S3, C5×S3, C8⋊C22, C5×D4, C22×C10, C2×D12, S3×C10, D4×C10, C8⋊D6, C5×D12, S3×C2×C10, C5×C8⋊C22, C10×D12, C5×C8⋊D6
►On 120 points
Generators in S120
(1 61 26 51 93)(2 62 27 52 94)(3 63 28 53 95)(4 64 29 54 96)(5 57 30 55 89)(6 58 31 56 90)(7 59 32 49 91)(8 60 25 50 92)(9 85 110 22 36)(10 86 111 23 37)(11 87 112 24 38)(12 88 105 17 39)(13 81 106 18 40)(14 82 107 19 33)(15 83 108 20 34)(16 84 109 21 35)(41 115 74 99 65)(42 116 75 100 66)(43 117 76 101 67)(44 118 77 102 68)(45 119 78 103 69)(46 120 79 104 70)(47 113 80 97 71)(48 114 73 98 72)
(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)
(1 103 18)(2 100 19 6 104 23)(3 97 20)(4 102 21 8 98 17)(5 99 22)(7 101 24)(9 30 41)(10 27 42 14 31 46)(11 32 43)(12 29 44 16 25 48)(13 26 45)(15 28 47)(33 58 70 37 62 66)(34 63 71)(35 60 72 39 64 68)(36 57 65)(38 59 67)(40 61 69)(49 117 87)(50 114 88 54 118 84)(51 119 81)(52 116 82 56 120 86)(53 113 83)(55 115 85)(73 105 96 77 109 92)(74 110 89)(75 107 90 79 111 94)(76 112 91)(78 106 93)(80 108 95)
(1 18)(2 17)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 32)(16 31)(33 60)(34 59)(35 58)(36 57)(37 64)(38 63)(39 62)(40 61)(42 48)(43 47)(44 46)(49 83)(50 82)(51 81)(52 88)(53 87)(54 86)(55 85)(56 84)(66 72)(67 71)(68 70)(73 75)(76 80)(77 79)(89 110)(90 109)(91 108)(92 107)(93 106)(94 105)(95 112)(96 111)(97 101)(98 100)(102 104)(113 117)(114 116)(118 120)
G:=sub<Sym(120)| (1,61,26,51,93)(2,62,27,52,94)(3,63,28,53,95)(4,64,29,54,96)(5,57,30,55,89)(6,58,31,56,90)(7,59,32,49,91)(8,60,25,50,92)(9,85,110,22,36)(10,86,111,23,37)(11,87,112,24,38)(12,88,105,17,39)(13,81,106,18,40)(14,82,107,19,33)(15,83,108,20,34)(16,84,109,21,35)(41,115,74,99,65)(42,116,75,100,66)(43,117,76,101,67)(44,118,77,102,68)(45,119,78,103,69)(46,120,79,104,70)(47,113,80,97,71)(48,114,73,98,72), (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), (1,103,18)(2,100,19,6,104,23)(3,97,20)(4,102,21,8,98,17)(5,99,22)(7,101,24)(9,30,41)(10,27,42,14,31,46)(11,32,43)(12,29,44,16,25,48)(13,26,45)(15,28,47)(33,58,70,37,62,66)(34,63,71)(35,60,72,39,64,68)(36,57,65)(38,59,67)(40,61,69)(49,117,87)(50,114,88,54,118,84)(51,119,81)(52,116,82,56,120,86)(53,113,83)(55,115,85)(73,105,96,77,109,92)(74,110,89)(75,107,90,79,111,94)(76,112,91)(78,106,93)(80,108,95), (1,18)(2,17)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,32)(16,31)(33,60)(34,59)(35,58)(36,57)(37,64)(38,63)(39,62)(40,61)(42,48)(43,47)(44,46)(49,83)(50,82)(51,81)(52,88)(53,87)(54,86)(55,85)(56,84)(66,72)(67,71)(68,70)(73,75)(76,80)(77,79)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,112)(96,111)(97,101)(98,100)(102,104)(113,117)(114,116)(118,120)>;
G:=Group( (1,61,26,51,93)(2,62,27,52,94)(3,63,28,53,95)(4,64,29,54,96)(5,57,30,55,89)(6,58,31,56,90)(7,59,32,49,91)(8,60,25,50,92)(9,85,110,22,36)(10,86,111,23,37)(11,87,112,24,38)(12,88,105,17,39)(13,81,106,18,40)(14,82,107,19,33)(15,83,108,20,34)(16,84,109,21,35)(41,115,74,99,65)(42,116,75,100,66)(43,117,76,101,67)(44,118,77,102,68)(45,119,78,103,69)(46,120,79,104,70)(47,113,80,97,71)(48,114,73,98,72), (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), (1,103,18)(2,100,19,6,104,23)(3,97,20)(4,102,21,8,98,17)(5,99,22)(7,101,24)(9,30,41)(10,27,42,14,31,46)(11,32,43)(12,29,44,16,25,48)(13,26,45)(15,28,47)(33,58,70,37,62,66)(34,63,71)(35,60,72,39,64,68)(36,57,65)(38,59,67)(40,61,69)(49,117,87)(50,114,88,54,118,84)(51,119,81)(52,116,82,56,120,86)(53,113,83)(55,115,85)(73,105,96,77,109,92)(74,110,89)(75,107,90,79,111,94)(76,112,91)(78,106,93)(80,108,95), (1,18)(2,17)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,32)(16,31)(33,60)(34,59)(35,58)(36,57)(37,64)(38,63)(39,62)(40,61)(42,48)(43,47)(44,46)(49,83)(50,82)(51,81)(52,88)(53,87)(54,86)(55,85)(56,84)(66,72)(67,71)(68,70)(73,75)(76,80)(77,79)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,112)(96,111)(97,101)(98,100)(102,104)(113,117)(114,116)(118,120) );
G=PermutationGroup([[(1,61,26,51,93),(2,62,27,52,94),(3,63,28,53,95),(4,64,29,54,96),(5,57,30,55,89),(6,58,31,56,90),(7,59,32,49,91),(8,60,25,50,92),(9,85,110,22,36),(10,86,111,23,37),(11,87,112,24,38),(12,88,105,17,39),(13,81,106,18,40),(14,82,107,19,33),(15,83,108,20,34),(16,84,109,21,35),(41,115,74,99,65),(42,116,75,100,66),(43,117,76,101,67),(44,118,77,102,68),(45,119,78,103,69),(46,120,79,104,70),(47,113,80,97,71),(48,114,73,98,72)], [(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)], [(1,103,18),(2,100,19,6,104,23),(3,97,20),(4,102,21,8,98,17),(5,99,22),(7,101,24),(9,30,41),(10,27,42,14,31,46),(11,32,43),(12,29,44,16,25,48),(13,26,45),(15,28,47),(33,58,70,37,62,66),(34,63,71),(35,60,72,39,64,68),(36,57,65),(38,59,67),(40,61,69),(49,117,87),(50,114,88,54,118,84),(51,119,81),(52,116,82,56,120,86),(53,113,83),(55,115,85),(73,105,96,77,109,92),(74,110,89),(75,107,90,79,111,94),(76,112,91),(78,106,93),(80,108,95)], [(1,18),(2,17),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,32),(16,31),(33,60),(34,59),(35,58),(36,57),(37,64),(38,63),(39,62),(40,61),(42,48),(43,47),(44,46),(49,83),(50,82),(51,81),(52,88),(53,87),(54,86),(55,85),(56,84),(66,72),(67,71),(68,70),(73,75),(76,80),(77,79),(89,110),(90,109),(91,108),(92,107),(93,106),(94,105),(95,112),(96,111),(97,101),(98,100),(102,104),(113,117),(114,116),(118,120)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10T | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 12 | 12 | 12 | 2 | 2 | 2 | 12 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
105 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 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | D12 | D12 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×D12 | C5×D12 | C8⋊C22 | C8⋊D6 | C5×C8⋊C22 | C5×C8⋊D6 |
| kernel | C5×C8⋊D6 | C5×C24⋊C2 | C5×D24 | C15×M4(2) | C10×D12 | C5×C4○D12 | C8⋊D6 | C24⋊C2 | D24 | C3×M4(2) | C2×D12 | C4○D12 | C5×M4(2) | C60 | C2×C30 | C40 | C2×C20 | C20 | C2×C10 | M4(2) | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 8 | 8 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C5×C8⋊D6 ►in GL4(𝔽241) generated by
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 99 | 43 | 0 | 0 |
| 198 | 142 | 0 | 0 |
| 240 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 43 | 99 |
| 0 | 0 | 142 | 198 |
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[0,0,99,198,0,0,43,142,1,0,0,0,0,1,0,0],[240,240,0,0,1,0,0,0,0,0,1,1,0,0,240,0],[0,1,0,0,1,0,0,0,0,0,43,142,0,0,99,198] >;
C5×C8⋊D6 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes D_6
% in TeX
G:=Group("C5xC8:D6"); // GroupNames label
G:=SmallGroup(480,787);
// by ID
G=gap.SmallGroup(480,787);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,891,226,4204,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^6=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations