direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D4.S3, C30.48D4, C20.37D6, C15⋊12SD16, Dic6⋊2C10, C60.44C22, C3⋊C8⋊2C10, D4.(C5×S3), C6.8(C5×D4), C3⋊2(C5×SD16), C4.2(S3×C10), (C5×D4).2S3, C12.2(C2×C10), (C5×Dic6)⋊8C2, (D4×C15).3C2, (C3×D4).1C10, C10.24(C3⋊D4), (C5×C3⋊C8)⋊9C2, C2.5(C5×C3⋊D4), SmallGroup(240,61)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4.S3
G = < a,b,c,d,e | a5=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Smallest permutation representation of C5×D4.S3
►On 120 points
Generators in S120
(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 26 58 66)(2 27 59 67)(3 28 60 68)(4 29 56 69)(5 30 57 70)(6 20 90 52)(7 16 86 53)(8 17 87 54)(9 18 88 55)(10 19 89 51)(11 98 48 111)(12 99 49 112)(13 100 50 113)(14 96 46 114)(15 97 47 115)(21 75 93 107)(22 71 94 108)(23 72 95 109)(24 73 91 110)(25 74 92 106)(31 76 43 63)(32 77 44 64)(33 78 45 65)(34 79 41 61)(35 80 42 62)(36 118 103 81)(37 119 104 82)(38 120 105 83)(39 116 101 84)(40 117 102 85)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 52)(7 53)(8 54)(9 55)(10 51)(16 86)(17 87)(18 88)(19 89)(20 90)(21 93)(22 94)(23 95)(24 91)(25 92)(26 58)(27 59)(28 60)(29 56)(30 57)(31 63)(32 64)(33 65)(34 61)(35 62)(41 79)(42 80)(43 76)(44 77)(45 78)(81 118)(82 119)(83 120)(84 116)(85 117)(96 114)(97 115)(98 111)(99 112)(100 113)
(1 33 90)(2 34 86)(3 35 87)(4 31 88)(5 32 89)(6 58 45)(7 59 41)(8 60 42)(9 56 43)(10 57 44)(11 36 73)(12 37 74)(13 38 75)(14 39 71)(15 40 72)(16 67 61)(17 68 62)(18 69 63)(19 70 64)(20 66 65)(21 113 83)(22 114 84)(23 115 85)(24 111 81)(25 112 82)(26 78 52)(27 79 53)(28 80 54)(29 76 55)(30 77 51)(46 101 108)(47 102 109)(48 103 110)(49 104 106)(50 105 107)(91 98 118)(92 99 119)(93 100 120)(94 96 116)(95 97 117)
(1 118 58 81)(2 119 59 82)(3 120 60 83)(4 116 56 84)(5 117 57 85)(6 24 90 91)(7 25 86 92)(8 21 87 93)(9 22 88 94)(10 23 89 95)(11 65 48 78)(12 61 49 79)(13 62 50 80)(14 63 46 76)(15 64 47 77)(16 106 53 74)(17 107 54 75)(18 108 55 71)(19 109 51 72)(20 110 52 73)(26 36 66 103)(27 37 67 104)(28 38 68 105)(29 39 69 101)(30 40 70 102)(31 96 43 114)(32 97 44 115)(33 98 45 111)(34 99 41 112)(35 100 42 113)
G:=sub<Sym(120)| (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), (1,26,58,66)(2,27,59,67)(3,28,60,68)(4,29,56,69)(5,30,57,70)(6,20,90,52)(7,16,86,53)(8,17,87,54)(9,18,88,55)(10,19,89,51)(11,98,48,111)(12,99,49,112)(13,100,50,113)(14,96,46,114)(15,97,47,115)(21,75,93,107)(22,71,94,108)(23,72,95,109)(24,73,91,110)(25,74,92,106)(31,76,43,63)(32,77,44,64)(33,78,45,65)(34,79,41,61)(35,80,42,62)(36,118,103,81)(37,119,104,82)(38,120,105,83)(39,116,101,84)(40,117,102,85), (1,66)(2,67)(3,68)(4,69)(5,70)(6,52)(7,53)(8,54)(9,55)(10,51)(16,86)(17,87)(18,88)(19,89)(20,90)(21,93)(22,94)(23,95)(24,91)(25,92)(26,58)(27,59)(28,60)(29,56)(30,57)(31,63)(32,64)(33,65)(34,61)(35,62)(41,79)(42,80)(43,76)(44,77)(45,78)(81,118)(82,119)(83,120)(84,116)(85,117)(96,114)(97,115)(98,111)(99,112)(100,113), (1,33,90)(2,34,86)(3,35,87)(4,31,88)(5,32,89)(6,58,45)(7,59,41)(8,60,42)(9,56,43)(10,57,44)(11,36,73)(12,37,74)(13,38,75)(14,39,71)(15,40,72)(16,67,61)(17,68,62)(18,69,63)(19,70,64)(20,66,65)(21,113,83)(22,114,84)(23,115,85)(24,111,81)(25,112,82)(26,78,52)(27,79,53)(28,80,54)(29,76,55)(30,77,51)(46,101,108)(47,102,109)(48,103,110)(49,104,106)(50,105,107)(91,98,118)(92,99,119)(93,100,120)(94,96,116)(95,97,117), (1,118,58,81)(2,119,59,82)(3,120,60,83)(4,116,56,84)(5,117,57,85)(6,24,90,91)(7,25,86,92)(8,21,87,93)(9,22,88,94)(10,23,89,95)(11,65,48,78)(12,61,49,79)(13,62,50,80)(14,63,46,76)(15,64,47,77)(16,106,53,74)(17,107,54,75)(18,108,55,71)(19,109,51,72)(20,110,52,73)(26,36,66,103)(27,37,67,104)(28,38,68,105)(29,39,69,101)(30,40,70,102)(31,96,43,114)(32,97,44,115)(33,98,45,111)(34,99,41,112)(35,100,42,113)>;
G:=Group( (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), (1,26,58,66)(2,27,59,67)(3,28,60,68)(4,29,56,69)(5,30,57,70)(6,20,90,52)(7,16,86,53)(8,17,87,54)(9,18,88,55)(10,19,89,51)(11,98,48,111)(12,99,49,112)(13,100,50,113)(14,96,46,114)(15,97,47,115)(21,75,93,107)(22,71,94,108)(23,72,95,109)(24,73,91,110)(25,74,92,106)(31,76,43,63)(32,77,44,64)(33,78,45,65)(34,79,41,61)(35,80,42,62)(36,118,103,81)(37,119,104,82)(38,120,105,83)(39,116,101,84)(40,117,102,85), (1,66)(2,67)(3,68)(4,69)(5,70)(6,52)(7,53)(8,54)(9,55)(10,51)(16,86)(17,87)(18,88)(19,89)(20,90)(21,93)(22,94)(23,95)(24,91)(25,92)(26,58)(27,59)(28,60)(29,56)(30,57)(31,63)(32,64)(33,65)(34,61)(35,62)(41,79)(42,80)(43,76)(44,77)(45,78)(81,118)(82,119)(83,120)(84,116)(85,117)(96,114)(97,115)(98,111)(99,112)(100,113), (1,33,90)(2,34,86)(3,35,87)(4,31,88)(5,32,89)(6,58,45)(7,59,41)(8,60,42)(9,56,43)(10,57,44)(11,36,73)(12,37,74)(13,38,75)(14,39,71)(15,40,72)(16,67,61)(17,68,62)(18,69,63)(19,70,64)(20,66,65)(21,113,83)(22,114,84)(23,115,85)(24,111,81)(25,112,82)(26,78,52)(27,79,53)(28,80,54)(29,76,55)(30,77,51)(46,101,108)(47,102,109)(48,103,110)(49,104,106)(50,105,107)(91,98,118)(92,99,119)(93,100,120)(94,96,116)(95,97,117), (1,118,58,81)(2,119,59,82)(3,120,60,83)(4,116,56,84)(5,117,57,85)(6,24,90,91)(7,25,86,92)(8,21,87,93)(9,22,88,94)(10,23,89,95)(11,65,48,78)(12,61,49,79)(13,62,50,80)(14,63,46,76)(15,64,47,77)(16,106,53,74)(17,107,54,75)(18,108,55,71)(19,109,51,72)(20,110,52,73)(26,36,66,103)(27,37,67,104)(28,38,68,105)(29,39,69,101)(30,40,70,102)(31,96,43,114)(32,97,44,115)(33,98,45,111)(34,99,41,112)(35,100,42,113) );
G=PermutationGroup([[(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)], [(1,26,58,66),(2,27,59,67),(3,28,60,68),(4,29,56,69),(5,30,57,70),(6,20,90,52),(7,16,86,53),(8,17,87,54),(9,18,88,55),(10,19,89,51),(11,98,48,111),(12,99,49,112),(13,100,50,113),(14,96,46,114),(15,97,47,115),(21,75,93,107),(22,71,94,108),(23,72,95,109),(24,73,91,110),(25,74,92,106),(31,76,43,63),(32,77,44,64),(33,78,45,65),(34,79,41,61),(35,80,42,62),(36,118,103,81),(37,119,104,82),(38,120,105,83),(39,116,101,84),(40,117,102,85)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,52),(7,53),(8,54),(9,55),(10,51),(16,86),(17,87),(18,88),(19,89),(20,90),(21,93),(22,94),(23,95),(24,91),(25,92),(26,58),(27,59),(28,60),(29,56),(30,57),(31,63),(32,64),(33,65),(34,61),(35,62),(41,79),(42,80),(43,76),(44,77),(45,78),(81,118),(82,119),(83,120),(84,116),(85,117),(96,114),(97,115),(98,111),(99,112),(100,113)], [(1,33,90),(2,34,86),(3,35,87),(4,31,88),(5,32,89),(6,58,45),(7,59,41),(8,60,42),(9,56,43),(10,57,44),(11,36,73),(12,37,74),(13,38,75),(14,39,71),(15,40,72),(16,67,61),(17,68,62),(18,69,63),(19,70,64),(20,66,65),(21,113,83),(22,114,84),(23,115,85),(24,111,81),(25,112,82),(26,78,52),(27,79,53),(28,80,54),(29,76,55),(30,77,51),(46,101,108),(47,102,109),(48,103,110),(49,104,106),(50,105,107),(91,98,118),(92,99,119),(93,100,120),(94,96,116),(95,97,117)], [(1,118,58,81),(2,119,59,82),(3,120,60,83),(4,116,56,84),(5,117,57,85),(6,24,90,91),(7,25,86,92),(8,21,87,93),(9,22,88,94),(10,23,89,95),(11,65,48,78),(12,61,49,79),(13,62,50,80),(14,63,46,76),(15,64,47,77),(16,106,53,74),(17,107,54,75),(18,108,55,71),(19,109,51,72),(20,110,52,73),(26,36,66,103),(27,37,67,104),(28,38,68,105),(29,39,69,101),(30,40,70,102),(31,96,43,114),(32,97,44,115),(33,98,45,111),(34,99,41,112),(35,100,42,113)]])
C5×D4.S3 is a maximal subgroup of
C60.8C23 Dic10⋊D6 D30.9D4 D20.9D6 C60.16C23 D20.10D6 Dic6⋊D10 C5×S3×SD16
60 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12 | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 40A | ··· | 40H | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 4 | 2 | 2 | 12 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 6 | 6 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | SD16 | C3⋊D4 | C5×S3 | C5×D4 | S3×C10 | C5×SD16 | C5×C3⋊D4 | D4.S3 | C5×D4.S3 |
| kernel | C5×D4.S3 | C5×C3⋊C8 | C5×Dic6 | D4×C15 | D4.S3 | C3⋊C8 | Dic6 | C3×D4 | C5×D4 | C30 | C20 | C15 | C10 | D4 | C6 | C4 | C3 | C2 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 4 |
Matrix representation of C5×D4.S3 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 91 |
| 1 | 227 | 0 | 0 |
| 69 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 227 | 0 | 0 |
| 0 | 240 | 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 |
| 0 | 25 | 0 | 0 |
| 106 | 0 | 0 | 0 |
| 0 | 0 | 109 | 221 |
| 0 | 0 | 112 | 132 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,91,0,0,0,0,91],[1,69,0,0,227,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,227,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,240],[0,106,0,0,25,0,0,0,0,0,109,112,0,0,221,132] >;
C5×D4.S3 in GAP, Magma, Sage, TeX
C_5\times D_4.S_3
% in TeX
G:=Group("C5xD4.S3"); // GroupNames label
G:=SmallGroup(240,61);
// by ID
G=gap.SmallGroup(240,61);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,240,265,1443,729,69,5765]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^3=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations
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