direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C5⋊2C8, C20.30D6, C12.30D10, D6.2Dic5, C60.30C22, Dic3.2Dic5, C5⋊5(S3×C8), C15⋊6(C2×C8), (C5×S3)⋊2C8, (C4×S3).3D5, C15⋊3C8⋊10C2, C4.23(S3×D5), (S3×C20).2C2, (S3×C10).3C4, C10.17(C4×S3), C30.23(C2×C4), C2.1(S3×Dic5), C6.1(C2×Dic5), (C5×Dic3).3C4, C3⋊1(C2×C5⋊2C8), (C3×C5⋊2C8)⋊4C2, SmallGroup(240,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — S3×C5⋊2C8 |
Generators and relations for S3×C5⋊2C8
G = < a,b,c,d | a3=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of S3×C5⋊2C8
►On 120 points
Generators in S120
(1 11 50)(2 12 51)(3 13 52)(4 14 53)(5 15 54)(6 16 55)(7 9 56)(8 10 49)(17 33 25)(18 34 26)(19 35 27)(20 36 28)(21 37 29)(22 38 30)(23 39 31)(24 40 32)(41 113 71)(42 114 72)(43 115 65)(44 116 66)(45 117 67)(46 118 68)(47 119 69)(48 120 70)(57 102 78)(58 103 79)(59 104 80)(60 97 73)(61 98 74)(62 99 75)(63 100 76)(64 101 77)(81 110 93)(82 111 94)(83 112 95)(84 105 96)(85 106 89)(86 107 90)(87 108 91)(88 109 92)
(1 5)(2 6)(3 7)(4 8)(9 52)(10 53)(11 54)(12 55)(13 56)(14 49)(15 50)(16 51)(17 21)(18 22)(19 23)(20 24)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 65)(48 66)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 73)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 105)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(113 117)(114 118)(115 119)(116 120)
(1 99 20 113 96)(2 89 114 21 100)(3 101 22 115 90)(4 91 116 23 102)(5 103 24 117 92)(6 93 118 17 104)(7 97 18 119 94)(8 95 120 19 98)(9 73 34 69 82)(10 83 70 35 74)(11 75 36 71 84)(12 85 72 37 76)(13 77 38 65 86)(14 87 66 39 78)(15 79 40 67 88)(16 81 68 33 80)(25 59 55 110 46)(26 47 111 56 60)(27 61 49 112 48)(28 41 105 50 62)(29 63 51 106 42)(30 43 107 52 64)(31 57 53 108 44)(32 45 109 54 58)
(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)
G:=sub<Sym(120)| (1,11,50)(2,12,51)(3,13,52)(4,14,53)(5,15,54)(6,16,55)(7,9,56)(8,10,49)(17,33,25)(18,34,26)(19,35,27)(20,36,28)(21,37,29)(22,38,30)(23,39,31)(24,40,32)(41,113,71)(42,114,72)(43,115,65)(44,116,66)(45,117,67)(46,118,68)(47,119,69)(48,120,70)(57,102,78)(58,103,79)(59,104,80)(60,97,73)(61,98,74)(62,99,75)(63,100,76)(64,101,77)(81,110,93)(82,111,94)(83,112,95)(84,105,96)(85,106,89)(86,107,90)(87,108,91)(88,109,92), (1,5)(2,6)(3,7)(4,8)(9,52)(10,53)(11,54)(12,55)(13,56)(14,49)(15,50)(16,51)(17,21)(18,22)(19,23)(20,24)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,65)(48,66)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,73)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,105)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(113,117)(114,118)(115,119)(116,120), (1,99,20,113,96)(2,89,114,21,100)(3,101,22,115,90)(4,91,116,23,102)(5,103,24,117,92)(6,93,118,17,104)(7,97,18,119,94)(8,95,120,19,98)(9,73,34,69,82)(10,83,70,35,74)(11,75,36,71,84)(12,85,72,37,76)(13,77,38,65,86)(14,87,66,39,78)(15,79,40,67,88)(16,81,68,33,80)(25,59,55,110,46)(26,47,111,56,60)(27,61,49,112,48)(28,41,105,50,62)(29,63,51,106,42)(30,43,107,52,64)(31,57,53,108,44)(32,45,109,54,58), (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)>;
G:=Group( (1,11,50)(2,12,51)(3,13,52)(4,14,53)(5,15,54)(6,16,55)(7,9,56)(8,10,49)(17,33,25)(18,34,26)(19,35,27)(20,36,28)(21,37,29)(22,38,30)(23,39,31)(24,40,32)(41,113,71)(42,114,72)(43,115,65)(44,116,66)(45,117,67)(46,118,68)(47,119,69)(48,120,70)(57,102,78)(58,103,79)(59,104,80)(60,97,73)(61,98,74)(62,99,75)(63,100,76)(64,101,77)(81,110,93)(82,111,94)(83,112,95)(84,105,96)(85,106,89)(86,107,90)(87,108,91)(88,109,92), (1,5)(2,6)(3,7)(4,8)(9,52)(10,53)(11,54)(12,55)(13,56)(14,49)(15,50)(16,51)(17,21)(18,22)(19,23)(20,24)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(32,36)(41,67)(42,68)(43,69)(44,70)(45,71)(46,72)(47,65)(48,66)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,73)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,105)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(113,117)(114,118)(115,119)(116,120), (1,99,20,113,96)(2,89,114,21,100)(3,101,22,115,90)(4,91,116,23,102)(5,103,24,117,92)(6,93,118,17,104)(7,97,18,119,94)(8,95,120,19,98)(9,73,34,69,82)(10,83,70,35,74)(11,75,36,71,84)(12,85,72,37,76)(13,77,38,65,86)(14,87,66,39,78)(15,79,40,67,88)(16,81,68,33,80)(25,59,55,110,46)(26,47,111,56,60)(27,61,49,112,48)(28,41,105,50,62)(29,63,51,106,42)(30,43,107,52,64)(31,57,53,108,44)(32,45,109,54,58), (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) );
G=PermutationGroup([[(1,11,50),(2,12,51),(3,13,52),(4,14,53),(5,15,54),(6,16,55),(7,9,56),(8,10,49),(17,33,25),(18,34,26),(19,35,27),(20,36,28),(21,37,29),(22,38,30),(23,39,31),(24,40,32),(41,113,71),(42,114,72),(43,115,65),(44,116,66),(45,117,67),(46,118,68),(47,119,69),(48,120,70),(57,102,78),(58,103,79),(59,104,80),(60,97,73),(61,98,74),(62,99,75),(63,100,76),(64,101,77),(81,110,93),(82,111,94),(83,112,95),(84,105,96),(85,106,89),(86,107,90),(87,108,91),(88,109,92)], [(1,5),(2,6),(3,7),(4,8),(9,52),(10,53),(11,54),(12,55),(13,56),(14,49),(15,50),(16,51),(17,21),(18,22),(19,23),(20,24),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(32,36),(41,67),(42,68),(43,69),(44,70),(45,71),(46,72),(47,65),(48,66),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,73),(81,106),(82,107),(83,108),(84,109),(85,110),(86,111),(87,112),(88,105),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(113,117),(114,118),(115,119),(116,120)], [(1,99,20,113,96),(2,89,114,21,100),(3,101,22,115,90),(4,91,116,23,102),(5,103,24,117,92),(6,93,118,17,104),(7,97,18,119,94),(8,95,120,19,98),(9,73,34,69,82),(10,83,70,35,74),(11,75,36,71,84),(12,85,72,37,76),(13,77,38,65,86),(14,87,66,39,78),(15,79,40,67,88),(16,81,68,33,80),(25,59,55,110,46),(26,47,111,56,60),(27,61,49,112,48),(28,41,105,50,62),(29,63,51,106,42),(30,43,107,52,64),(31,57,53,108,44),(32,45,109,54,58)], [(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)]])
S3×C5⋊2C8 is a maximal subgroup of
C15⋊M5(2) S3×C8×D5 C40.34D6 C40.35D6 D12.2Dic5 D12.Dic5 D20.24D6 C60.19C23 D20.27D6 Dic10.27D6
S3×C5⋊2C8 is a maximal quotient of
C40.52D6 C60.94D4 C60.15Q8
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 15 | 15 | 15 | 15 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D5 | D6 | Dic5 | D10 | Dic5 | C4×S3 | C5⋊2C8 | S3×C8 | S3×D5 | S3×Dic5 | S3×C5⋊2C8 |
| kernel | S3×C5⋊2C8 | C3×C5⋊2C8 | C15⋊3C8 | S3×C20 | C5×Dic3 | S3×C10 | C5×S3 | C5⋊2C8 | C4×S3 | C20 | Dic3 | C12 | D6 | C10 | S3 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 8 | 4 | 2 | 2 | 4 |
Matrix representation of S3×C5⋊2C8 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 240 |
| 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 51 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 24 | 174 | 0 | 0 |
| 193 | 217 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,240,240],[240,0,0,0,0,240,0,0,0,0,0,1,0,0,1,0],[51,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[24,193,0,0,174,217,0,0,0,0,240,0,0,0,0,240] >;
S3×C5⋊2C8 in GAP, Magma, Sage, TeX
S_3\times C_5\rtimes_2C_8
% in TeX
G:=Group("S3xC5:2C8"); // GroupNames label
G:=SmallGroup(240,8);
// by ID
G=gap.SmallGroup(240,8);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^5=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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