direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×S3×D8, C40⋊26D6, D24⋊4C10, C120⋊27C22, C60.217C23, C3⋊2(C10×D8), C8⋊4(S3×C10), C15⋊15(C2×D8), (S3×C8)⋊1C10, C24⋊2(C2×C10), D4⋊S3⋊1C10, (S3×D4)⋊1C10, (C5×D4)⋊16D6, D4⋊1(S3×C10), (C3×D8)⋊2C10, (C15×D8)⋊9C2, (S3×C40)⋊10C2, D12⋊1(C2×C10), (C5×D24)⋊12C2, D6.12(C5×D4), C6.27(D4×C10), (S3×C10).48D4, C10.181(S3×D4), C30.363(C2×D4), Dic3.3(C5×D4), (C5×D12)⋊18C22, (D4×C15)⋊18C22, C12.1(C22×C10), (C5×Dic3).30D4, (S3×C20).56C22, C20.190(C22×S3), (C5×S3×D4)⋊8C2, C3⋊C8⋊5(C2×C10), C4.1(S3×C2×C10), C2.15(C5×S3×D4), (C5×D4⋊S3)⋊9C2, (C3×D4)⋊1(C2×C10), (C5×C3⋊C8)⋊38C22, (C4×S3).7(C2×C10), SmallGroup(480,789)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×S3×D8
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=d-1 >
Subgroups: 484 in 152 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C8, C2×C4, D4, D4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, D8, D8, C2×D4, C20, C20, C2×C10, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C5×S3, C5×S3, C30, C30, C2×D8, C40, C40, C2×C20, C5×D4, C5×D4, C22×C10, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C40, C5×D8, C5×D8, D4×C10, S3×D8, C5×C3⋊C8, C120, S3×C20, C5×D12, C5×C3⋊D4, D4×C15, S3×C2×C10, C10×D8, S3×C40, C5×D24, C5×D4⋊S3, C15×D8, C5×S3×D4, C5×S3×D8
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, D8, C2×D4, C2×C10, C22×S3, C5×S3, C2×D8, C5×D4, C22×C10, S3×D4, S3×C10, C5×D8, D4×C10, S3×D8, S3×C2×C10, C10×D8, C5×S3×D4, C5×S3×D8
►On 120 points
Generators in S120
(1 62 90 52 74)(2 63 91 53 75)(3 64 92 54 76)(4 57 93 55 77)(5 58 94 56 78)(6 59 95 49 79)(7 60 96 50 80)(8 61 89 51 73)(9 29 110 22 45)(10 30 111 23 46)(11 31 112 24 47)(12 32 105 17 48)(13 25 106 18 41)(14 26 107 19 42)(15 27 108 20 43)(16 28 109 21 44)(33 119 82 103 66)(34 120 83 104 67)(35 113 84 97 68)(36 114 85 98 69)(37 115 86 99 70)(38 116 87 100 71)(39 117 88 101 72)(40 118 81 102 65)
(1 22 99)(2 23 100)(3 24 101)(4 17 102)(5 18 103)(6 19 104)(7 20 97)(8 21 98)(9 37 90)(10 38 91)(11 39 92)(12 40 93)(13 33 94)(14 34 95)(15 35 96)(16 36 89)(25 119 56)(26 120 49)(27 113 50)(28 114 51)(29 115 52)(30 116 53)(31 117 54)(32 118 55)(41 66 58)(42 67 59)(43 68 60)(44 69 61)(45 70 62)(46 71 63)(47 72 64)(48 65 57)(73 109 85)(74 110 86)(75 111 87)(76 112 88)(77 105 81)(78 106 82)(79 107 83)(80 108 84)
(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(25 119)(26 120)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 65)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)
(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)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 16)(14 15)(17 22)(18 21)(19 20)(23 24)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)(37 40)(38 39)(41 44)(42 43)(45 48)(46 47)(49 50)(51 56)(52 55)(53 54)(57 62)(58 61)(59 60)(63 64)(65 70)(66 69)(67 68)(71 72)(73 78)(74 77)(75 76)(79 80)(81 86)(82 85)(83 84)(87 88)(89 94)(90 93)(91 92)(95 96)(97 104)(98 103)(99 102)(100 101)(105 110)(106 109)(107 108)(111 112)(113 120)(114 119)(115 118)(116 117)
G:=sub<Sym(120)| (1,62,90,52,74)(2,63,91,53,75)(3,64,92,54,76)(4,57,93,55,77)(5,58,94,56,78)(6,59,95,49,79)(7,60,96,50,80)(8,61,89,51,73)(9,29,110,22,45)(10,30,111,23,46)(11,31,112,24,47)(12,32,105,17,48)(13,25,106,18,41)(14,26,107,19,42)(15,27,108,20,43)(16,28,109,21,44)(33,119,82,103,66)(34,120,83,104,67)(35,113,84,97,68)(36,114,85,98,69)(37,115,86,99,70)(38,116,87,100,71)(39,117,88,101,72)(40,118,81,102,65), (1,22,99)(2,23,100)(3,24,101)(4,17,102)(5,18,103)(6,19,104)(7,20,97)(8,21,98)(9,37,90)(10,38,91)(11,39,92)(12,40,93)(13,33,94)(14,34,95)(15,35,96)(16,36,89)(25,119,56)(26,120,49)(27,113,50)(28,114,51)(29,115,52)(30,116,53)(31,117,54)(32,118,55)(41,66,58)(42,67,59)(43,68,60)(44,69,61)(45,70,62)(46,71,63)(47,72,64)(48,65,57)(73,109,85)(74,110,86)(75,111,87)(76,112,88)(77,105,81)(78,106,82)(79,107,83)(80,108,84), (9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,102)(18,103)(19,104)(20,97)(21,98)(22,99)(23,100)(24,101)(25,119)(26,120)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112), (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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,22)(18,21)(19,20)(23,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)(45,48)(46,47)(49,50)(51,56)(52,55)(53,54)(57,62)(58,61)(59,60)(63,64)(65,70)(66,69)(67,68)(71,72)(73,78)(74,77)(75,76)(79,80)(81,86)(82,85)(83,84)(87,88)(89,94)(90,93)(91,92)(95,96)(97,104)(98,103)(99,102)(100,101)(105,110)(106,109)(107,108)(111,112)(113,120)(114,119)(115,118)(116,117)>;
G:=Group( (1,62,90,52,74)(2,63,91,53,75)(3,64,92,54,76)(4,57,93,55,77)(5,58,94,56,78)(6,59,95,49,79)(7,60,96,50,80)(8,61,89,51,73)(9,29,110,22,45)(10,30,111,23,46)(11,31,112,24,47)(12,32,105,17,48)(13,25,106,18,41)(14,26,107,19,42)(15,27,108,20,43)(16,28,109,21,44)(33,119,82,103,66)(34,120,83,104,67)(35,113,84,97,68)(36,114,85,98,69)(37,115,86,99,70)(38,116,87,100,71)(39,117,88,101,72)(40,118,81,102,65), (1,22,99)(2,23,100)(3,24,101)(4,17,102)(5,18,103)(6,19,104)(7,20,97)(8,21,98)(9,37,90)(10,38,91)(11,39,92)(12,40,93)(13,33,94)(14,34,95)(15,35,96)(16,36,89)(25,119,56)(26,120,49)(27,113,50)(28,114,51)(29,115,52)(30,116,53)(31,117,54)(32,118,55)(41,66,58)(42,67,59)(43,68,60)(44,69,61)(45,70,62)(46,71,63)(47,72,64)(48,65,57)(73,109,85)(74,110,86)(75,111,87)(76,112,88)(77,105,81)(78,106,82)(79,107,83)(80,108,84), (9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,102)(18,103)(19,104)(20,97)(21,98)(22,99)(23,100)(24,101)(25,119)(26,120)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112), (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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,22)(18,21)(19,20)(23,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)(45,48)(46,47)(49,50)(51,56)(52,55)(53,54)(57,62)(58,61)(59,60)(63,64)(65,70)(66,69)(67,68)(71,72)(73,78)(74,77)(75,76)(79,80)(81,86)(82,85)(83,84)(87,88)(89,94)(90,93)(91,92)(95,96)(97,104)(98,103)(99,102)(100,101)(105,110)(106,109)(107,108)(111,112)(113,120)(114,119)(115,118)(116,117) );
G=PermutationGroup([[(1,62,90,52,74),(2,63,91,53,75),(3,64,92,54,76),(4,57,93,55,77),(5,58,94,56,78),(6,59,95,49,79),(7,60,96,50,80),(8,61,89,51,73),(9,29,110,22,45),(10,30,111,23,46),(11,31,112,24,47),(12,32,105,17,48),(13,25,106,18,41),(14,26,107,19,42),(15,27,108,20,43),(16,28,109,21,44),(33,119,82,103,66),(34,120,83,104,67),(35,113,84,97,68),(36,114,85,98,69),(37,115,86,99,70),(38,116,87,100,71),(39,117,88,101,72),(40,118,81,102,65)], [(1,22,99),(2,23,100),(3,24,101),(4,17,102),(5,18,103),(6,19,104),(7,20,97),(8,21,98),(9,37,90),(10,38,91),(11,39,92),(12,40,93),(13,33,94),(14,34,95),(15,35,96),(16,36,89),(25,119,56),(26,120,49),(27,113,50),(28,114,51),(29,115,52),(30,116,53),(31,117,54),(32,118,55),(41,66,58),(42,67,59),(43,68,60),(44,69,61),(45,70,62),(46,71,63),(47,72,64),(48,65,57),(73,109,85),(74,110,86),(75,111,87),(76,112,88),(77,105,81),(78,106,82),(79,107,83),(80,108,84)], [(9,37),(10,38),(11,39),(12,40),(13,33),(14,34),(15,35),(16,36),(17,102),(18,103),(19,104),(20,97),(21,98),(22,99),(23,100),(24,101),(25,119),(26,120),(27,113),(28,114),(29,115),(30,116),(31,117),(32,118),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,65),(81,105),(82,106),(83,107),(84,108),(85,109),(86,110),(87,111),(88,112)], [(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)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,16),(14,15),(17,22),(18,21),(19,20),(23,24),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35),(37,40),(38,39),(41,44),(42,43),(45,48),(46,47),(49,50),(51,56),(52,55),(53,54),(57,62),(58,61),(59,60),(63,64),(65,70),(66,69),(67,68),(71,72),(73,78),(74,77),(75,76),(79,80),(81,86),(82,85),(83,84),(87,88),(89,94),(90,93),(91,92),(95,96),(97,104),(98,103),(99,102),(100,101),(105,110),(106,109),(107,108),(111,112),(113,120),(114,119),(115,118),(116,117)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 12 | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 40A | ··· | 40H | 40I | ··· | 40P | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 12 | 12 | 2 | 2 | 6 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 6 | ··· | 6 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D4 | D6 | D6 | D8 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | S3×C10 | C5×D8 | S3×D4 | S3×D8 | C5×S3×D4 | C5×S3×D8 |
| kernel | C5×S3×D8 | S3×C40 | C5×D24 | C5×D4⋊S3 | C15×D8 | C5×S3×D4 | S3×D8 | S3×C8 | D24 | D4⋊S3 | C3×D8 | S3×D4 | C5×D8 | C5×Dic3 | S3×C10 | C40 | C5×D4 | C5×S3 | D8 | Dic3 | D6 | C8 | D4 | S3 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C5×S3×D8 ►in GL4(𝔽241) generated by
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 91 |
| 0 | 240 | 0 | 0 |
| 1 | 240 | 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 | 230 | 11 |
| 0 | 0 | 230 | 230 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 11 | 230 |
| 0 | 0 | 230 | 230 |
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,91,0,0,0,0,91],[0,1,0,0,240,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,240,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,230,230,0,0,11,230],[240,0,0,0,0,240,0,0,0,0,11,230,0,0,230,230] >;
C5×S3×D8 in GAP, Magma, Sage, TeX
C_5\times S_3\times D_8
% in TeX
G:=Group("C5xS3xD8"); // GroupNames label
G:=SmallGroup(480,789);
// by ID
G=gap.SmallGroup(480,789);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,471,2111,1068,102,15686]);
// 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^-1>;
// generators/relations