direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C8⋊S3, C40⋊7S3, D6.C20, C24⋊5C10, C120⋊13C2, C20.57D6, Dic3.C20, C15⋊12M4(2), C60.74C22, C3⋊C8⋊4C10, C8⋊3(C5×S3), C2.3(S3×C20), C6.2(C2×C20), C3⋊1(C5×M4(2)), (C4×S3).2C10, (S3×C20).5C2, (S3×C10).5C4, C10.23(C4×S3), C4.13(S3×C10), C30.46(C2×C4), C12.13(C2×C10), (C5×Dic3).5C4, (C5×C3⋊C8)⋊11C2, SmallGroup(240,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊S3
G = < a,b,c,d | a5=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Smallest permutation representation of C5×C8⋊S3
►On 120 points
Generators in S120
(1 61 26 51 93)(2 62 27 52 94)(3 63 28 53 95)(4 64 29 54 96)(5 57 30 55 89)(6 58 31 56 90)(7 59 32 49 91)(8 60 25 50 92)(9 85 110 22 36)(10 86 111 23 37)(11 87 112 24 38)(12 88 105 17 39)(13 81 106 18 40)(14 82 107 19 33)(15 83 108 20 34)(16 84 109 21 35)(41 115 74 99 65)(42 116 75 100 66)(43 117 76 101 67)(44 118 77 102 68)(45 119 78 103 69)(46 120 79 104 70)(47 113 80 97 71)(48 114 73 98 72)
(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)
(1 103 18)(2 104 19)(3 97 20)(4 98 21)(5 99 22)(6 100 23)(7 101 24)(8 102 17)(9 30 41)(10 31 42)(11 32 43)(12 25 44)(13 26 45)(14 27 46)(15 28 47)(16 29 48)(33 62 70)(34 63 71)(35 64 72)(36 57 65)(37 58 66)(38 59 67)(39 60 68)(40 61 69)(49 117 87)(50 118 88)(51 119 81)(52 120 82)(53 113 83)(54 114 84)(55 115 85)(56 116 86)(73 109 96)(74 110 89)(75 111 90)(76 112 91)(77 105 92)(78 106 93)(79 107 94)(80 108 95)
(2 6)(4 8)(9 41)(10 46)(11 43)(12 48)(13 45)(14 42)(15 47)(16 44)(17 98)(18 103)(19 100)(20 97)(21 102)(22 99)(23 104)(24 101)(25 29)(27 31)(33 66)(34 71)(35 68)(36 65)(37 70)(38 67)(39 72)(40 69)(50 54)(52 56)(58 62)(60 64)(73 105)(74 110)(75 107)(76 112)(77 109)(78 106)(79 111)(80 108)(81 119)(82 116)(83 113)(84 118)(85 115)(86 120)(87 117)(88 114)(90 94)(92 96)
G:=sub<Sym(120)| (1,61,26,51,93)(2,62,27,52,94)(3,63,28,53,95)(4,64,29,54,96)(5,57,30,55,89)(6,58,31,56,90)(7,59,32,49,91)(8,60,25,50,92)(9,85,110,22,36)(10,86,111,23,37)(11,87,112,24,38)(12,88,105,17,39)(13,81,106,18,40)(14,82,107,19,33)(15,83,108,20,34)(16,84,109,21,35)(41,115,74,99,65)(42,116,75,100,66)(43,117,76,101,67)(44,118,77,102,68)(45,119,78,103,69)(46,120,79,104,70)(47,113,80,97,71)(48,114,73,98,72), (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), (1,103,18)(2,104,19)(3,97,20)(4,98,21)(5,99,22)(6,100,23)(7,101,24)(8,102,17)(9,30,41)(10,31,42)(11,32,43)(12,25,44)(13,26,45)(14,27,46)(15,28,47)(16,29,48)(33,62,70)(34,63,71)(35,64,72)(36,57,65)(37,58,66)(38,59,67)(39,60,68)(40,61,69)(49,117,87)(50,118,88)(51,119,81)(52,120,82)(53,113,83)(54,114,84)(55,115,85)(56,116,86)(73,109,96)(74,110,89)(75,111,90)(76,112,91)(77,105,92)(78,106,93)(79,107,94)(80,108,95), (2,6)(4,8)(9,41)(10,46)(11,43)(12,48)(13,45)(14,42)(15,47)(16,44)(17,98)(18,103)(19,100)(20,97)(21,102)(22,99)(23,104)(24,101)(25,29)(27,31)(33,66)(34,71)(35,68)(36,65)(37,70)(38,67)(39,72)(40,69)(50,54)(52,56)(58,62)(60,64)(73,105)(74,110)(75,107)(76,112)(77,109)(78,106)(79,111)(80,108)(81,119)(82,116)(83,113)(84,118)(85,115)(86,120)(87,117)(88,114)(90,94)(92,96)>;
G:=Group( (1,61,26,51,93)(2,62,27,52,94)(3,63,28,53,95)(4,64,29,54,96)(5,57,30,55,89)(6,58,31,56,90)(7,59,32,49,91)(8,60,25,50,92)(9,85,110,22,36)(10,86,111,23,37)(11,87,112,24,38)(12,88,105,17,39)(13,81,106,18,40)(14,82,107,19,33)(15,83,108,20,34)(16,84,109,21,35)(41,115,74,99,65)(42,116,75,100,66)(43,117,76,101,67)(44,118,77,102,68)(45,119,78,103,69)(46,120,79,104,70)(47,113,80,97,71)(48,114,73,98,72), (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), (1,103,18)(2,104,19)(3,97,20)(4,98,21)(5,99,22)(6,100,23)(7,101,24)(8,102,17)(9,30,41)(10,31,42)(11,32,43)(12,25,44)(13,26,45)(14,27,46)(15,28,47)(16,29,48)(33,62,70)(34,63,71)(35,64,72)(36,57,65)(37,58,66)(38,59,67)(39,60,68)(40,61,69)(49,117,87)(50,118,88)(51,119,81)(52,120,82)(53,113,83)(54,114,84)(55,115,85)(56,116,86)(73,109,96)(74,110,89)(75,111,90)(76,112,91)(77,105,92)(78,106,93)(79,107,94)(80,108,95), (2,6)(4,8)(9,41)(10,46)(11,43)(12,48)(13,45)(14,42)(15,47)(16,44)(17,98)(18,103)(19,100)(20,97)(21,102)(22,99)(23,104)(24,101)(25,29)(27,31)(33,66)(34,71)(35,68)(36,65)(37,70)(38,67)(39,72)(40,69)(50,54)(52,56)(58,62)(60,64)(73,105)(74,110)(75,107)(76,112)(77,109)(78,106)(79,111)(80,108)(81,119)(82,116)(83,113)(84,118)(85,115)(86,120)(87,117)(88,114)(90,94)(92,96) );
G=PermutationGroup([[(1,61,26,51,93),(2,62,27,52,94),(3,63,28,53,95),(4,64,29,54,96),(5,57,30,55,89),(6,58,31,56,90),(7,59,32,49,91),(8,60,25,50,92),(9,85,110,22,36),(10,86,111,23,37),(11,87,112,24,38),(12,88,105,17,39),(13,81,106,18,40),(14,82,107,19,33),(15,83,108,20,34),(16,84,109,21,35),(41,115,74,99,65),(42,116,75,100,66),(43,117,76,101,67),(44,118,77,102,68),(45,119,78,103,69),(46,120,79,104,70),(47,113,80,97,71),(48,114,73,98,72)], [(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)], [(1,103,18),(2,104,19),(3,97,20),(4,98,21),(5,99,22),(6,100,23),(7,101,24),(8,102,17),(9,30,41),(10,31,42),(11,32,43),(12,25,44),(13,26,45),(14,27,46),(15,28,47),(16,29,48),(33,62,70),(34,63,71),(35,64,72),(36,57,65),(37,58,66),(38,59,67),(39,60,68),(40,61,69),(49,117,87),(50,118,88),(51,119,81),(52,120,82),(53,113,83),(54,114,84),(55,115,85),(56,116,86),(73,109,96),(74,110,89),(75,111,90),(76,112,91),(77,105,92),(78,106,93),(79,107,94),(80,108,95)], [(2,6),(4,8),(9,41),(10,46),(11,43),(12,48),(13,45),(14,42),(15,47),(16,44),(17,98),(18,103),(19,100),(20,97),(21,102),(22,99),(23,104),(24,101),(25,29),(27,31),(33,66),(34,71),(35,68),(36,65),(37,70),(38,67),(39,72),(40,69),(50,54),(52,56),(58,62),(60,64),(73,105),(74,110),(75,107),(76,112),(77,109),(78,106),(79,111),(80,108),(81,119),(82,116),(83,113),(84,118),(85,115),(86,120),(87,117),(88,114),(90,94),(92,96)]])
C5×C8⋊S3 is a maximal subgroup of
C40⋊D6 C40⋊1D6 D40⋊S3 Dic20⋊S3 C40.34D6 C40.35D6 C40.2D6 C5×S3×M4(2)
90 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 40I | ··· | 40P | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 6 | 2 | 1 | 1 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
90 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 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | S3 | D6 | M4(2) | C4×S3 | C5×S3 | C8⋊S3 | S3×C10 | C5×M4(2) | S3×C20 | C5×C8⋊S3 |
| kernel | C5×C8⋊S3 | C5×C3⋊C8 | C120 | S3×C20 | C5×Dic3 | S3×C10 | C8⋊S3 | C3⋊C8 | C24 | C4×S3 | Dic3 | D6 | C40 | C20 | C15 | C10 | C8 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C5×C8⋊S3 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 120 | 0 | 0 |
| 113 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,98,0,0,0,0,98],[0,113,0,0,120,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,240],[1,0,0,0,0,240,0,0,0,0,0,1,0,0,1,0] >;
C5×C8⋊S3 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes S_3
% in TeX
G:=Group("C5xC8:S3"); // GroupNames label
G:=SmallGroup(240,50);
// by ID
G=gap.SmallGroup(240,50);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,505,127,69,5765]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations
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