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, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, SD16, SD16, C2×D4, C2×Q8, C20, C20, C2×C10, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C5×S3, C30, C30, C2×SD16, C40, C40, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, 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, C2×C30, C2×C40, C5×SD16, C5×SD16, 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, C22, C5, S3, D4, C23, C10, D6, SD16, C2×D4, C2×C10, C22×S3, C5×S3, C2×SD16, C5×D4, C22×C10, S3×D4, S3×C10, C5×SD16, D4×C10, S3×SD16, S3×C2×C10, C10×SD16, C5×S3×D4, C5×S3×SD16
►On 120 points
Generators in S120
(1 58 31 56 88)(2 59 32 49 81)(3 60 25 50 82)(4 61 26 51 83)(5 62 27 52 84)(6 63 28 53 85)(7 64 29 54 86)(8 57 30 55 87)(9 46 110 18 40)(10 47 111 19 33)(11 48 112 20 34)(12 41 105 21 35)(13 42 106 22 36)(14 43 107 23 37)(15 44 108 24 38)(16 45 109 17 39)(65 94 115 78 99)(66 95 116 79 100)(67 96 117 80 101)(68 89 118 73 102)(69 90 119 74 103)(70 91 120 75 104)(71 92 113 76 97)(72 93 114 77 98)
(1 22 103)(2 23 104)(3 24 97)(4 17 98)(5 18 99)(6 19 100)(7 20 101)(8 21 102)(9 94 27)(10 95 28)(11 96 29)(12 89 30)(13 90 31)(14 91 32)(15 92 25)(16 93 26)(33 66 63)(34 67 64)(35 68 57)(36 69 58)(37 70 59)(38 71 60)(39 72 61)(40 65 62)(41 118 55)(42 119 56)(43 120 49)(44 113 50)(45 114 51)(46 115 52)(47 116 53)(48 117 54)(73 87 105)(74 88 106)(75 81 107)(76 82 108)(77 83 109)(78 84 110)(79 85 111)(80 86 112)
(9 94)(10 95)(11 96)(12 89)(13 90)(14 91)(15 92)(16 93)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 97)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 65)(41 118)(42 119)(43 120)(44 113)(45 114)(46 115)(47 116)(48 117)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)
(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 23)(19 21)(20 24)(25 29)(26 32)(28 30)(33 35)(34 38)(37 39)(41 47)(43 45)(44 48)(49 51)(50 54)(53 55)(57 63)(59 61)(60 64)(66 68)(67 71)(70 72)(73 79)(75 77)(76 80)(81 83)(82 86)(85 87)(89 95)(91 93)(92 96)(97 101)(98 104)(100 102)(105 111)(107 109)(108 112)(113 117)(114 120)(116 118)
G:=sub<Sym(120)| (1,58,31,56,88)(2,59,32,49,81)(3,60,25,50,82)(4,61,26,51,83)(5,62,27,52,84)(6,63,28,53,85)(7,64,29,54,86)(8,57,30,55,87)(9,46,110,18,40)(10,47,111,19,33)(11,48,112,20,34)(12,41,105,21,35)(13,42,106,22,36)(14,43,107,23,37)(15,44,108,24,38)(16,45,109,17,39)(65,94,115,78,99)(66,95,116,79,100)(67,96,117,80,101)(68,89,118,73,102)(69,90,119,74,103)(70,91,120,75,104)(71,92,113,76,97)(72,93,114,77,98), (1,22,103)(2,23,104)(3,24,97)(4,17,98)(5,18,99)(6,19,100)(7,20,101)(8,21,102)(9,94,27)(10,95,28)(11,96,29)(12,89,30)(13,90,31)(14,91,32)(15,92,25)(16,93,26)(33,66,63)(34,67,64)(35,68,57)(36,69,58)(37,70,59)(38,71,60)(39,72,61)(40,65,62)(41,118,55)(42,119,56)(43,120,49)(44,113,50)(45,114,51)(46,115,52)(47,116,53)(48,117,54)(73,87,105)(74,88,106)(75,81,107)(76,82,108)(77,83,109)(78,84,110)(79,85,111)(80,86,112), (9,94)(10,95)(11,96)(12,89)(13,90)(14,91)(15,92)(16,93)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,97)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,65)(41,118)(42,119)(43,120)(44,113)(45,114)(46,115)(47,116)(48,117)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112), (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)(17,23)(19,21)(20,24)(25,29)(26,32)(28,30)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)(49,51)(50,54)(53,55)(57,63)(59,61)(60,64)(66,68)(67,71)(70,72)(73,79)(75,77)(76,80)(81,83)(82,86)(85,87)(89,95)(91,93)(92,96)(97,101)(98,104)(100,102)(105,111)(107,109)(108,112)(113,117)(114,120)(116,118)>;
G:=Group( (1,58,31,56,88)(2,59,32,49,81)(3,60,25,50,82)(4,61,26,51,83)(5,62,27,52,84)(6,63,28,53,85)(7,64,29,54,86)(8,57,30,55,87)(9,46,110,18,40)(10,47,111,19,33)(11,48,112,20,34)(12,41,105,21,35)(13,42,106,22,36)(14,43,107,23,37)(15,44,108,24,38)(16,45,109,17,39)(65,94,115,78,99)(66,95,116,79,100)(67,96,117,80,101)(68,89,118,73,102)(69,90,119,74,103)(70,91,120,75,104)(71,92,113,76,97)(72,93,114,77,98), (1,22,103)(2,23,104)(3,24,97)(4,17,98)(5,18,99)(6,19,100)(7,20,101)(8,21,102)(9,94,27)(10,95,28)(11,96,29)(12,89,30)(13,90,31)(14,91,32)(15,92,25)(16,93,26)(33,66,63)(34,67,64)(35,68,57)(36,69,58)(37,70,59)(38,71,60)(39,72,61)(40,65,62)(41,118,55)(42,119,56)(43,120,49)(44,113,50)(45,114,51)(46,115,52)(47,116,53)(48,117,54)(73,87,105)(74,88,106)(75,81,107)(76,82,108)(77,83,109)(78,84,110)(79,85,111)(80,86,112), (9,94)(10,95)(11,96)(12,89)(13,90)(14,91)(15,92)(16,93)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,97)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,65)(41,118)(42,119)(43,120)(44,113)(45,114)(46,115)(47,116)(48,117)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112), (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)(17,23)(19,21)(20,24)(25,29)(26,32)(28,30)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)(49,51)(50,54)(53,55)(57,63)(59,61)(60,64)(66,68)(67,71)(70,72)(73,79)(75,77)(76,80)(81,83)(82,86)(85,87)(89,95)(91,93)(92,96)(97,101)(98,104)(100,102)(105,111)(107,109)(108,112)(113,117)(114,120)(116,118) );
G=PermutationGroup([[(1,58,31,56,88),(2,59,32,49,81),(3,60,25,50,82),(4,61,26,51,83),(5,62,27,52,84),(6,63,28,53,85),(7,64,29,54,86),(8,57,30,55,87),(9,46,110,18,40),(10,47,111,19,33),(11,48,112,20,34),(12,41,105,21,35),(13,42,106,22,36),(14,43,107,23,37),(15,44,108,24,38),(16,45,109,17,39),(65,94,115,78,99),(66,95,116,79,100),(67,96,117,80,101),(68,89,118,73,102),(69,90,119,74,103),(70,91,120,75,104),(71,92,113,76,97),(72,93,114,77,98)], [(1,22,103),(2,23,104),(3,24,97),(4,17,98),(5,18,99),(6,19,100),(7,20,101),(8,21,102),(9,94,27),(10,95,28),(11,96,29),(12,89,30),(13,90,31),(14,91,32),(15,92,25),(16,93,26),(33,66,63),(34,67,64),(35,68,57),(36,69,58),(37,70,59),(38,71,60),(39,72,61),(40,65,62),(41,118,55),(42,119,56),(43,120,49),(44,113,50),(45,114,51),(46,115,52),(47,116,53),(48,117,54),(73,87,105),(74,88,106),(75,81,107),(76,82,108),(77,83,109),(78,84,110),(79,85,111),(80,86,112)], [(9,94),(10,95),(11,96),(12,89),(13,90),(14,91),(15,92),(16,93),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,104),(24,97),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,65),(41,118),(42,119),(43,120),(44,113),(45,114),(46,115),(47,116),(48,117),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112)], [(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),(17,23),(19,21),(20,24),(25,29),(26,32),(28,30),(33,35),(34,38),(37,39),(41,47),(43,45),(44,48),(49,51),(50,54),(53,55),(57,63),(59,61),(60,64),(66,68),(67,71),(70,72),(73,79),(75,77),(76,80),(81,83),(82,86),(85,87),(89,95),(91,93),(92,96),(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