direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Q8⋊3D6, C40⋊21D6, D24⋊6C10, C120⋊28C22, C60.221C23, C8⋊3(S3×C10), C24⋊3(C2×C10), D4⋊S3⋊3C10, (S3×D4)⋊3C10, (C5×Q8)⋊18D6, Q8⋊3(S3×C10), D6.7(C5×D4), C8⋊S3⋊1C10, (C5×D24)⋊14C2, D12⋊2(C2×C10), (C5×SD16)⋊5S3, SD16⋊1(C5×S3), (C5×D4).27D6, D4.3(S3×C10), C6.31(D4×C10), C15⋊33(C8⋊C22), Q8⋊2S3⋊2C10, Q8⋊3S3⋊1C10, (C15×SD16)⋊7C2, (C3×SD16)⋊1C10, (S3×C10).43D4, C10.185(S3×D4), C30.367(C2×D4), Dic3.9(C5×D4), (C5×D12)⋊19C22, C12.5(C22×C10), (C5×Dic3).46D4, (Q8×C15)⋊17C22, (S3×C20).37C22, C20.194(C22×S3), (D4×C15).32C22, C3⋊C8⋊2(C2×C10), (C5×S3×D4)⋊10C2, C3⋊3(C5×C8⋊C22), C4.5(S3×C2×C10), C2.19(C5×S3×D4), (C5×C8⋊S3)⋊9C2, (C5×D4⋊S3)⋊11C2, (C5×C3⋊C8)⋊24C22, (C3×Q8)⋊2(C2×C10), (C4×S3).2(C2×C10), (C5×Q8⋊3S3)⋊8C2, (C3×D4).3(C2×C10), (C5×Q8⋊2S3)⋊10C2, SmallGroup(480,793)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8⋊3D6
G = < a,b,c,d,e | a5=b4=d6=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >
Subgroups: 404 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, M4(2), D8, SD16, SD16, C2×D4, C4○D4, C20, C20, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C30, C30, C8⋊C22, C40, C40, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, C5×M4(2), C5×D8, C5×SD16, C5×SD16, D4×C10, C5×C4○D4, Q8⋊3D6, C5×C3⋊C8, C120, S3×C20, S3×C20, C5×D12, C5×D12, C5×C3⋊D4, D4×C15, Q8×C15, S3×C2×C10, C5×C8⋊C22, C5×C8⋊S3, C5×D24, C5×D4⋊S3, C5×Q8⋊2S3, C15×SD16, C5×S3×D4, C5×Q8⋊3S3, C5×Q8⋊3D6
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C22×S3, C5×S3, C8⋊C22, C5×D4, C22×C10, S3×D4, S3×C10, D4×C10, Q8⋊3D6, S3×C2×C10, C5×C8⋊C22, C5×S3×D4, C5×Q8⋊3D6
►On 120 points
Generators in S120
(1 30 6 21 15)(2 28 4 19 13)(3 29 5 20 14)(7 11 18 23 25)(8 12 16 24 26)(9 10 17 22 27)(31 74 103 79 112)(32 75 104 80 113)(33 76 105 81 114)(34 77 106 82 109)(35 78 107 83 110)(36 73 108 84 111)(37 43 116 91 53)(38 44 117 92 54)(39 45 118 93 49)(40 46 119 94 50)(41 47 120 95 51)(42 48 115 96 52)(55 62 71 97 85)(56 63 72 98 86)(57 64 67 99 87)(58 65 68 100 88)(59 66 69 101 89)(60 61 70 102 90)
(1 89 9 86)(2 87 7 90)(3 85 8 88)(4 64 18 61)(5 62 16 65)(6 66 17 63)(10 56 30 59)(11 60 28 57)(12 58 29 55)(13 99 25 102)(14 97 26 100)(15 101 27 98)(19 67 23 70)(20 71 24 68)(21 69 22 72)(31 34 40 37)(32 38 41 35)(33 36 42 39)(43 74 77 46)(44 47 78 75)(45 76 73 48)(49 114 111 52)(50 53 112 109)(51 110 113 54)(79 82 94 91)(80 92 95 83)(81 84 96 93)(103 106 119 116)(104 117 120 107)(105 108 115 118)
(1 96 9 81)(2 94 7 79)(3 92 8 83)(4 40 18 31)(5 38 16 35)(6 42 17 33)(10 114 30 52)(11 112 28 50)(12 110 29 54)(13 119 25 103)(14 117 26 107)(15 115 27 105)(19 46 23 74)(20 44 24 78)(21 48 22 76)(32 65 41 62)(34 61 37 64)(36 63 39 66)(43 67 77 70)(45 69 73 72)(47 71 75 68)(49 59 111 56)(51 55 113 58)(53 57 109 60)(80 88 95 85)(82 90 91 87)(84 86 93 89)(97 104 100 120)(98 118 101 108)(99 106 102 116)
(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 8)(2 7)(3 9)(4 18)(5 17)(6 16)(10 29)(11 28)(12 30)(13 25)(14 27)(15 26)(19 23)(20 22)(21 24)(31 34)(32 33)(35 36)(37 40)(38 39)(41 42)(43 46)(44 45)(47 48)(49 54)(50 53)(51 52)(55 59)(56 58)(62 66)(63 65)(68 72)(69 71)(73 78)(74 77)(75 76)(79 82)(80 81)(83 84)(85 89)(86 88)(91 94)(92 93)(95 96)(97 101)(98 100)(103 106)(104 105)(107 108)(109 112)(110 111)(113 114)(115 120)(116 119)(117 118)
G:=sub<Sym(120)| (1,30,6,21,15)(2,28,4,19,13)(3,29,5,20,14)(7,11,18,23,25)(8,12,16,24,26)(9,10,17,22,27)(31,74,103,79,112)(32,75,104,80,113)(33,76,105,81,114)(34,77,106,82,109)(35,78,107,83,110)(36,73,108,84,111)(37,43,116,91,53)(38,44,117,92,54)(39,45,118,93,49)(40,46,119,94,50)(41,47,120,95,51)(42,48,115,96,52)(55,62,71,97,85)(56,63,72,98,86)(57,64,67,99,87)(58,65,68,100,88)(59,66,69,101,89)(60,61,70,102,90), (1,89,9,86)(2,87,7,90)(3,85,8,88)(4,64,18,61)(5,62,16,65)(6,66,17,63)(10,56,30,59)(11,60,28,57)(12,58,29,55)(13,99,25,102)(14,97,26,100)(15,101,27,98)(19,67,23,70)(20,71,24,68)(21,69,22,72)(31,34,40,37)(32,38,41,35)(33,36,42,39)(43,74,77,46)(44,47,78,75)(45,76,73,48)(49,114,111,52)(50,53,112,109)(51,110,113,54)(79,82,94,91)(80,92,95,83)(81,84,96,93)(103,106,119,116)(104,117,120,107)(105,108,115,118), (1,96,9,81)(2,94,7,79)(3,92,8,83)(4,40,18,31)(5,38,16,35)(6,42,17,33)(10,114,30,52)(11,112,28,50)(12,110,29,54)(13,119,25,103)(14,117,26,107)(15,115,27,105)(19,46,23,74)(20,44,24,78)(21,48,22,76)(32,65,41,62)(34,61,37,64)(36,63,39,66)(43,67,77,70)(45,69,73,72)(47,71,75,68)(49,59,111,56)(51,55,113,58)(53,57,109,60)(80,88,95,85)(82,90,91,87)(84,86,93,89)(97,104,100,120)(98,118,101,108)(99,106,102,116), (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), (1,8)(2,7)(3,9)(4,18)(5,17)(6,16)(10,29)(11,28)(12,30)(13,25)(14,27)(15,26)(19,23)(20,22)(21,24)(31,34)(32,33)(35,36)(37,40)(38,39)(41,42)(43,46)(44,45)(47,48)(49,54)(50,53)(51,52)(55,59)(56,58)(62,66)(63,65)(68,72)(69,71)(73,78)(74,77)(75,76)(79,82)(80,81)(83,84)(85,89)(86,88)(91,94)(92,93)(95,96)(97,101)(98,100)(103,106)(104,105)(107,108)(109,112)(110,111)(113,114)(115,120)(116,119)(117,118)>;
G:=Group( (1,30,6,21,15)(2,28,4,19,13)(3,29,5,20,14)(7,11,18,23,25)(8,12,16,24,26)(9,10,17,22,27)(31,74,103,79,112)(32,75,104,80,113)(33,76,105,81,114)(34,77,106,82,109)(35,78,107,83,110)(36,73,108,84,111)(37,43,116,91,53)(38,44,117,92,54)(39,45,118,93,49)(40,46,119,94,50)(41,47,120,95,51)(42,48,115,96,52)(55,62,71,97,85)(56,63,72,98,86)(57,64,67,99,87)(58,65,68,100,88)(59,66,69,101,89)(60,61,70,102,90), (1,89,9,86)(2,87,7,90)(3,85,8,88)(4,64,18,61)(5,62,16,65)(6,66,17,63)(10,56,30,59)(11,60,28,57)(12,58,29,55)(13,99,25,102)(14,97,26,100)(15,101,27,98)(19,67,23,70)(20,71,24,68)(21,69,22,72)(31,34,40,37)(32,38,41,35)(33,36,42,39)(43,74,77,46)(44,47,78,75)(45,76,73,48)(49,114,111,52)(50,53,112,109)(51,110,113,54)(79,82,94,91)(80,92,95,83)(81,84,96,93)(103,106,119,116)(104,117,120,107)(105,108,115,118), (1,96,9,81)(2,94,7,79)(3,92,8,83)(4,40,18,31)(5,38,16,35)(6,42,17,33)(10,114,30,52)(11,112,28,50)(12,110,29,54)(13,119,25,103)(14,117,26,107)(15,115,27,105)(19,46,23,74)(20,44,24,78)(21,48,22,76)(32,65,41,62)(34,61,37,64)(36,63,39,66)(43,67,77,70)(45,69,73,72)(47,71,75,68)(49,59,111,56)(51,55,113,58)(53,57,109,60)(80,88,95,85)(82,90,91,87)(84,86,93,89)(97,104,100,120)(98,118,101,108)(99,106,102,116), (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), (1,8)(2,7)(3,9)(4,18)(5,17)(6,16)(10,29)(11,28)(12,30)(13,25)(14,27)(15,26)(19,23)(20,22)(21,24)(31,34)(32,33)(35,36)(37,40)(38,39)(41,42)(43,46)(44,45)(47,48)(49,54)(50,53)(51,52)(55,59)(56,58)(62,66)(63,65)(68,72)(69,71)(73,78)(74,77)(75,76)(79,82)(80,81)(83,84)(85,89)(86,88)(91,94)(92,93)(95,96)(97,101)(98,100)(103,106)(104,105)(107,108)(109,112)(110,111)(113,114)(115,120)(116,119)(117,118) );
G=PermutationGroup([[(1,30,6,21,15),(2,28,4,19,13),(3,29,5,20,14),(7,11,18,23,25),(8,12,16,24,26),(9,10,17,22,27),(31,74,103,79,112),(32,75,104,80,113),(33,76,105,81,114),(34,77,106,82,109),(35,78,107,83,110),(36,73,108,84,111),(37,43,116,91,53),(38,44,117,92,54),(39,45,118,93,49),(40,46,119,94,50),(41,47,120,95,51),(42,48,115,96,52),(55,62,71,97,85),(56,63,72,98,86),(57,64,67,99,87),(58,65,68,100,88),(59,66,69,101,89),(60,61,70,102,90)], [(1,89,9,86),(2,87,7,90),(3,85,8,88),(4,64,18,61),(5,62,16,65),(6,66,17,63),(10,56,30,59),(11,60,28,57),(12,58,29,55),(13,99,25,102),(14,97,26,100),(15,101,27,98),(19,67,23,70),(20,71,24,68),(21,69,22,72),(31,34,40,37),(32,38,41,35),(33,36,42,39),(43,74,77,46),(44,47,78,75),(45,76,73,48),(49,114,111,52),(50,53,112,109),(51,110,113,54),(79,82,94,91),(80,92,95,83),(81,84,96,93),(103,106,119,116),(104,117,120,107),(105,108,115,118)], [(1,96,9,81),(2,94,7,79),(3,92,8,83),(4,40,18,31),(5,38,16,35),(6,42,17,33),(10,114,30,52),(11,112,28,50),(12,110,29,54),(13,119,25,103),(14,117,26,107),(15,115,27,105),(19,46,23,74),(20,44,24,78),(21,48,22,76),(32,65,41,62),(34,61,37,64),(36,63,39,66),(43,67,77,70),(45,69,73,72),(47,71,75,68),(49,59,111,56),(51,55,113,58),(53,57,109,60),(80,88,95,85),(82,90,91,87),(84,86,93,89),(97,104,100,120),(98,118,101,108),(99,106,102,116)], [(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)], [(1,8),(2,7),(3,9),(4,18),(5,17),(6,16),(10,29),(11,28),(12,30),(13,25),(14,27),(15,26),(19,23),(20,22),(21,24),(31,34),(32,33),(35,36),(37,40),(38,39),(41,42),(43,46),(44,45),(47,48),(49,54),(50,53),(51,52),(55,59),(56,58),(62,66),(63,65),(68,72),(69,71),(73,78),(74,77),(75,76),(79,82),(80,81),(83,84),(85,89),(86,88),(91,94),(92,93),(95,96),(97,101),(98,100),(103,106),(104,105),(107,108),(109,112),(110,111),(113,114),(115,120),(116,119),(117,118)]])
90 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 | 10J | 10K | 10L | 10M | ··· | 10T | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 20K | 20L | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | 40B | 40C | 40D | 40E | 40F | 40G | 40H | 60A | 60B | 60C | 60D | 60E | 60F | 60G | 60H | 120A | ··· | 120H |
| 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 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 6 | 12 | 12 | 2 | 2 | 4 | 6 | 1 | 1 | 1 | 1 | 2 | 8 | 4 | 12 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | D6 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | S3×C10 | C8⋊C22 | S3×D4 | Q8⋊3D6 | C5×C8⋊C22 | C5×S3×D4 | C5×Q8⋊3D6 |
| kernel | C5×Q8⋊3D6 | C5×C8⋊S3 | C5×D24 | C5×D4⋊S3 | C5×Q8⋊2S3 | C15×SD16 | C5×S3×D4 | C5×Q8⋊3S3 | Q8⋊3D6 | C8⋊S3 | D24 | D4⋊S3 | Q8⋊2S3 | C3×SD16 | S3×D4 | Q8⋊3S3 | C5×SD16 | C5×Dic3 | S3×C10 | C40 | C5×D4 | C5×Q8 | SD16 | Dic3 | D6 | C8 | D4 | Q8 | C15 | C10 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C5×Q8⋊3D6 ►in GL4(𝔽241) generated by
| 205 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 |
| 0 | 0 | 205 | 0 |
| 0 | 0 | 0 | 205 |
| 240 | 0 | 240 | 0 |
| 0 | 240 | 0 | 240 |
| 2 | 0 | 1 | 0 |
| 0 | 2 | 0 | 1 |
| 0 | 0 | 116 | 9 |
| 0 | 0 | 232 | 125 |
| 232 | 18 | 0 | 0 |
| 223 | 9 | 0 | 0 |
| 240 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 2 | 239 | 1 | 240 |
| 2 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 239 | 0 | 240 |
| 239 | 0 | 240 | 0 |
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,205,0,0,0,0,205],[240,0,2,0,0,240,0,2,240,0,1,0,0,240,0,1],[0,0,232,223,0,0,18,9,116,232,0,0,9,125,0,0],[240,240,2,2,1,0,239,0,0,0,1,1,0,0,240,0],[0,1,0,239,1,0,239,0,0,0,0,240,0,0,240,0] >;
C5×Q8⋊3D6 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes_3D_6
% in TeX
G:=Group("C5xQ8:3D6"); // GroupNames label
G:=SmallGroup(480,793);
// by ID
G=gap.SmallGroup(480,793);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1766,471,436,2111,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^6=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations