direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×S3×SD16, C40⋊27D6, C120⋊32C22, C60.220C23, C8⋊5(S3×C10), (S3×D4).C10, (S3×C8)⋊4C10, C24⋊5(C2×C10), (S3×Q8)⋊1C10, Q8⋊2(S3×C10), (C5×Q8)⋊17D6, C3⋊2(C10×SD16), (S3×C40)⋊13C2, C24⋊C2⋊5C10, D4.S3⋊3C10, (C5×D4).26D6, D4.2(S3×C10), D6.13(C5×D4), C6.30(D4×C10), C15⋊21(C2×SD16), Q8⋊2S3⋊1C10, (C3×SD16)⋊3C10, Dic6⋊2(C2×C10), (S3×C10).49D4, D12.2(C2×C10), C10.184(S3×D4), C30.366(C2×D4), Dic3.4(C5×D4), (C15×SD16)⋊11C2, C12.4(C22×C10), (C5×Dic3).31D4, (Q8×C15)⋊16C22, (S3×C20).58C22, C20.193(C22×S3), (C5×Dic6)⋊17C22, (C5×D12).31C22, (D4×C15).31C22, (C5×S3×Q8)⋊8C2, C3⋊C8⋊6(C2×C10), C4.4(S3×C2×C10), (C5×S3×D4).2C2, C2.18(C5×S3×D4), (C5×C3⋊C8)⋊39C22, (C3×Q8)⋊1(C2×C10), (C5×C24⋊C2)⋊13C2, (C4×S3).9(C2×C10), (C5×D4.S3)⋊11C2, (C5×Q8⋊2S3)⋊9C2, (C3×D4).2(C2×C10), SmallGroup(480,792)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×S3×SD16
G = < a,b,c,d,e | a5=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >
Subgroups: 372 in 136 conjugacy classes, 58 normal (54 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×2], S3, C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, C10, C10 [×4], Dic3, Dic3, C12, C12, D6, D6 [×3], C2×C6, C15, C2×C8, SD16, SD16 [×3], C2×D4, C2×Q8, C20, C20 [×3], C2×C10 [×5], C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3 [×2], C5×S3, C30, C30, C2×SD16, C40, C40, C2×C20 [×2], C5×D4, C5×D4 [×2], C5×Q8, C5×Q8 [×2], C22×C10, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C3×SD16, S3×D4, S3×Q8, C5×Dic3, C5×Dic3, C60, C60, S3×C10, S3×C10 [×3], C2×C30, C2×C40, C5×SD16, C5×SD16 [×3], D4×C10, Q8×C10, S3×SD16, C5×C3⋊C8, C120, C5×Dic6, C5×Dic6, S3×C20, S3×C20, C5×D12, C5×C3⋊D4, D4×C15, Q8×C15, S3×C2×C10, C10×SD16, S3×C40, C5×C24⋊C2, C5×D4.S3, C5×Q8⋊2S3, C15×SD16, C5×S3×D4, C5×S3×Q8, C5×S3×SD16
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], SD16 [×2], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C2×SD16, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], C5×SD16 [×2], D4×C10, S3×SD16, S3×C2×C10, C10×SD16, C5×S3×D4, C5×S3×SD16
►On 120 points
Generators in S120
(1 61 91 51 29)(2 62 92 52 30)(3 63 93 53 31)(4 64 94 54 32)(5 57 95 55 25)(6 58 96 56 26)(7 59 89 49 27)(8 60 90 50 28)(9 81 110 20 40)(10 82 111 21 33)(11 83 112 22 34)(12 84 105 23 35)(13 85 106 24 36)(14 86 107 17 37)(15 87 108 18 38)(16 88 109 19 39)(41 114 77 98 72)(42 115 78 99 65)(43 116 79 100 66)(44 117 80 101 67)(45 118 73 102 68)(46 119 74 103 69)(47 120 75 104 70)(48 113 76 97 71)
(1 24 103)(2 17 104)(3 18 97)(4 19 98)(5 20 99)(6 21 100)(7 22 101)(8 23 102)(9 42 95)(10 43 96)(11 44 89)(12 45 90)(13 46 91)(14 47 92)(15 48 93)(16 41 94)(25 110 78)(26 111 79)(27 112 80)(28 105 73)(29 106 74)(30 107 75)(31 108 76)(32 109 77)(33 66 58)(34 67 59)(35 68 60)(36 69 61)(37 70 62)(38 71 63)(39 72 64)(40 65 57)(49 83 117)(50 84 118)(51 85 119)(52 86 120)(53 87 113)(54 88 114)(55 81 115)(56 82 116)
(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 104)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 65)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 113)(88 114)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 19)(18 22)(21 23)(26 28)(27 31)(30 32)(33 35)(34 38)(37 39)(41 47)(43 45)(44 48)(49 53)(50 56)(52 54)(58 60)(59 63)(62 64)(66 68)(67 71)(70 72)(73 79)(75 77)(76 80)(82 84)(83 87)(86 88)(89 93)(90 96)(92 94)(97 101)(98 104)(100 102)(105 111)(107 109)(108 112)(113 117)(114 120)(116 118)
G:=sub<Sym(120)| (1,61,91,51,29)(2,62,92,52,30)(3,63,93,53,31)(4,64,94,54,32)(5,57,95,55,25)(6,58,96,56,26)(7,59,89,49,27)(8,60,90,50,28)(9,81,110,20,40)(10,82,111,21,33)(11,83,112,22,34)(12,84,105,23,35)(13,85,106,24,36)(14,86,107,17,37)(15,87,108,18,38)(16,88,109,19,39)(41,114,77,98,72)(42,115,78,99,65)(43,116,79,100,66)(44,117,80,101,67)(45,118,73,102,68)(46,119,74,103,69)(47,120,75,104,70)(48,113,76,97,71), (1,24,103)(2,17,104)(3,18,97)(4,19,98)(5,20,99)(6,21,100)(7,22,101)(8,23,102)(9,42,95)(10,43,96)(11,44,89)(12,45,90)(13,46,91)(14,47,92)(15,48,93)(16,41,94)(25,110,78)(26,111,79)(27,112,80)(28,105,73)(29,106,74)(30,107,75)(31,108,76)(32,109,77)(33,66,58)(34,67,59)(35,68,60)(36,69,61)(37,70,62)(38,71,63)(39,72,64)(40,65,57)(49,83,117)(50,84,118)(51,85,119)(52,86,120)(53,87,113)(54,88,114)(55,81,115)(56,82,116), (9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,104)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,65)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,113)(88,114), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,19)(18,22)(21,23)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)(49,53)(50,56)(52,54)(58,60)(59,63)(62,64)(66,68)(67,71)(70,72)(73,79)(75,77)(76,80)(82,84)(83,87)(86,88)(89,93)(90,96)(92,94)(97,101)(98,104)(100,102)(105,111)(107,109)(108,112)(113,117)(114,120)(116,118)>;
G:=Group( (1,61,91,51,29)(2,62,92,52,30)(3,63,93,53,31)(4,64,94,54,32)(5,57,95,55,25)(6,58,96,56,26)(7,59,89,49,27)(8,60,90,50,28)(9,81,110,20,40)(10,82,111,21,33)(11,83,112,22,34)(12,84,105,23,35)(13,85,106,24,36)(14,86,107,17,37)(15,87,108,18,38)(16,88,109,19,39)(41,114,77,98,72)(42,115,78,99,65)(43,116,79,100,66)(44,117,80,101,67)(45,118,73,102,68)(46,119,74,103,69)(47,120,75,104,70)(48,113,76,97,71), (1,24,103)(2,17,104)(3,18,97)(4,19,98)(5,20,99)(6,21,100)(7,22,101)(8,23,102)(9,42,95)(10,43,96)(11,44,89)(12,45,90)(13,46,91)(14,47,92)(15,48,93)(16,41,94)(25,110,78)(26,111,79)(27,112,80)(28,105,73)(29,106,74)(30,107,75)(31,108,76)(32,109,77)(33,66,58)(34,67,59)(35,68,60)(36,69,61)(37,70,62)(38,71,63)(39,72,64)(40,65,57)(49,83,117)(50,84,118)(51,85,119)(52,86,120)(53,87,113)(54,88,114)(55,81,115)(56,82,116), (9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,41)(17,104)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,65)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,113)(88,114), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,19)(18,22)(21,23)(26,28)(27,31)(30,32)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)(49,53)(50,56)(52,54)(58,60)(59,63)(62,64)(66,68)(67,71)(70,72)(73,79)(75,77)(76,80)(82,84)(83,87)(86,88)(89,93)(90,96)(92,94)(97,101)(98,104)(100,102)(105,111)(107,109)(108,112)(113,117)(114,120)(116,118) );
G=PermutationGroup([(1,61,91,51,29),(2,62,92,52,30),(3,63,93,53,31),(4,64,94,54,32),(5,57,95,55,25),(6,58,96,56,26),(7,59,89,49,27),(8,60,90,50,28),(9,81,110,20,40),(10,82,111,21,33),(11,83,112,22,34),(12,84,105,23,35),(13,85,106,24,36),(14,86,107,17,37),(15,87,108,18,38),(16,88,109,19,39),(41,114,77,98,72),(42,115,78,99,65),(43,116,79,100,66),(44,117,80,101,67),(45,118,73,102,68),(46,119,74,103,69),(47,120,75,104,70),(48,113,76,97,71)], [(1,24,103),(2,17,104),(3,18,97),(4,19,98),(5,20,99),(6,21,100),(7,22,101),(8,23,102),(9,42,95),(10,43,96),(11,44,89),(12,45,90),(13,46,91),(14,47,92),(15,48,93),(16,41,94),(25,110,78),(26,111,79),(27,112,80),(28,105,73),(29,106,74),(30,107,75),(31,108,76),(32,109,77),(33,66,58),(34,67,59),(35,68,60),(36,69,61),(37,70,62),(38,71,63),(39,72,64),(40,65,57),(49,83,117),(50,84,118),(51,85,119),(52,86,120),(53,87,113),(54,88,114),(55,81,115),(56,82,116)], [(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,41),(17,104),(18,97),(19,98),(20,99),(21,100),(22,101),(23,102),(24,103),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,65),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,115),(82,116),(83,117),(84,118),(85,119),(86,120),(87,113),(88,114)], [(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)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(17,19),(18,22),(21,23),(26,28),(27,31),(30,32),(33,35),(34,38),(37,39),(41,47),(43,45),(44,48),(49,53),(50,56),(52,54),(58,60),(59,63),(62,64),(66,68),(67,71),(70,72),(73,79),(75,77),(76,80),(82,84),(83,87),(86,88),(89,93),(90,96),(92,94),(97,101),(98,104),(100,102),(105,111),(107,109),(108,112),(113,117),(114,120),(116,118)])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 10M | 10N | 10O | 10P | 10Q | 10R | 10S | 10T | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 20K | 20L | 20M | 20N | 20O | 20P | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 40I | ··· | 40P | 60A | 60B | 60C | 60D | 60E | 60F | 60G | 60H | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 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 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 3 | 3 | 4 | 12 | 2 | 2 | 4 | 6 | 12 | 1 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
105 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 | 2 | 2 | 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 | SD16 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | S3×C10 | C5×SD16 | S3×D4 | S3×SD16 | C5×S3×D4 | C5×S3×SD16 |
| kernel | C5×S3×SD16 | S3×C40 | C5×C24⋊C2 | C5×D4.S3 | C5×Q8⋊2S3 | C15×SD16 | C5×S3×D4 | C5×S3×Q8 | S3×SD16 | S3×C8 | C24⋊C2 | D4.S3 | Q8⋊2S3 | C3×SD16 | S3×D4 | S3×Q8 | C5×SD16 | C5×Dic3 | S3×C10 | C40 | C5×D4 | C5×Q8 | C5×S3 | SD16 | Dic3 | D6 | C8 | D4 | Q8 | S3 | C10 | C5 | 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 | 4 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C5×S3×SD16 ►in GL4(𝔽241) generated by
| 205 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 |
| 0 | 0 | 205 | 0 |
| 0 | 0 | 0 | 205 |
| 240 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 222 | 19 |
| 0 | 0 | 222 | 222 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,205,0,0,0,0,205],[240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,222,222,0,0,19,222],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
C5×S3×SD16 in GAP, Magma, Sage, TeX
C_5\times S_3\times {\rm SD}_{16} % in TeX
G:=Group("C5xS3xSD16"); // GroupNames label
G:=SmallGroup(480,792);
// by ID
G=gap.SmallGroup(480,792);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,471,436,2111,1068,102,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations