direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C4.Dic3, C60.11C4, C12.1C20, C20.59D6, C15⋊15M4(2), C20.7Dic3, C60.76C22, C3⋊C8⋊5C10, C4.(C5×Dic3), (C2×C20).8S3, C6.6(C2×C20), (C2×C6).3C20, C3⋊2(C5×M4(2)), C4.15(S3×C10), C30.59(C2×C4), (C2×C60).15C2, (C2×C30).11C4, (C2×C12).5C10, C12.15(C2×C10), C22.(C5×Dic3), (C2×C10).3Dic3, C2.3(C10×Dic3), C10.19(C2×Dic3), (C5×C3⋊C8)⋊12C2, (C2×C4).2(C5×S3), SmallGroup(240,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4.Dic3
G = < a,b,c,d | a5=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Smallest permutation representation of C5×C4.Dic3
►On 120 points
Generators in S120
(1 49 37 31 22)(2 50 38 32 23)(3 51 39 33 24)(4 52 40 34 13)(5 53 41 35 14)(6 54 42 36 15)(7 55 43 25 16)(8 56 44 26 17)(9 57 45 27 18)(10 58 46 28 19)(11 59 47 29 20)(12 60 48 30 21)(61 112 98 86 75)(62 113 99 87 76)(63 114 100 88 77)(64 115 101 89 78)(65 116 102 90 79)(66 117 103 91 80)(67 118 104 92 81)(68 119 105 93 82)(69 120 106 94 83)(70 109 107 95 84)(71 110 108 96 73)(72 111 97 85 74)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 100 103 106)(98 101 104 107)(99 102 105 108)(109 112 115 118)(110 113 116 119)(111 114 117 120)
(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 70 10 67 7 64 4 61)(2 63 11 72 8 69 5 66)(3 68 12 65 9 62 6 71)(13 75 22 84 19 81 16 78)(14 80 23 77 20 74 17 83)(15 73 24 82 21 79 18 76)(25 89 34 86 31 95 28 92)(26 94 35 91 32 88 29 85)(27 87 36 96 33 93 30 90)(37 107 46 104 43 101 40 98)(38 100 47 97 44 106 41 103)(39 105 48 102 45 99 42 108)(49 109 58 118 55 115 52 112)(50 114 59 111 56 120 53 117)(51 119 60 116 57 113 54 110)
G:=sub<Sym(120)| (1,49,37,31,22)(2,50,38,32,23)(3,51,39,33,24)(4,52,40,34,13)(5,53,41,35,14)(6,54,42,36,15)(7,55,43,25,16)(8,56,44,26,17)(9,57,45,27,18)(10,58,46,28,19)(11,59,47,29,20)(12,60,48,30,21)(61,112,98,86,75)(62,113,99,87,76)(63,114,100,88,77)(64,115,101,89,78)(65,116,102,90,79)(66,117,103,91,80)(67,118,104,92,81)(68,119,105,93,82)(69,120,106,94,83)(70,109,107,95,84)(71,110,108,96,73)(72,111,97,85,74), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,112,115,118)(110,113,116,119)(111,114,117,120), (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), (1,70,10,67,7,64,4,61)(2,63,11,72,8,69,5,66)(3,68,12,65,9,62,6,71)(13,75,22,84,19,81,16,78)(14,80,23,77,20,74,17,83)(15,73,24,82,21,79,18,76)(25,89,34,86,31,95,28,92)(26,94,35,91,32,88,29,85)(27,87,36,96,33,93,30,90)(37,107,46,104,43,101,40,98)(38,100,47,97,44,106,41,103)(39,105,48,102,45,99,42,108)(49,109,58,118,55,115,52,112)(50,114,59,111,56,120,53,117)(51,119,60,116,57,113,54,110)>;
G:=Group( (1,49,37,31,22)(2,50,38,32,23)(3,51,39,33,24)(4,52,40,34,13)(5,53,41,35,14)(6,54,42,36,15)(7,55,43,25,16)(8,56,44,26,17)(9,57,45,27,18)(10,58,46,28,19)(11,59,47,29,20)(12,60,48,30,21)(61,112,98,86,75)(62,113,99,87,76)(63,114,100,88,77)(64,115,101,89,78)(65,116,102,90,79)(66,117,103,91,80)(67,118,104,92,81)(68,119,105,93,82)(69,120,106,94,83)(70,109,107,95,84)(71,110,108,96,73)(72,111,97,85,74), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,46,43,40)(38,47,44,41)(39,48,45,42)(49,58,55,52)(50,59,56,53)(51,60,57,54)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,88,91,94)(86,89,92,95)(87,90,93,96)(97,100,103,106)(98,101,104,107)(99,102,105,108)(109,112,115,118)(110,113,116,119)(111,114,117,120), (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), (1,70,10,67,7,64,4,61)(2,63,11,72,8,69,5,66)(3,68,12,65,9,62,6,71)(13,75,22,84,19,81,16,78)(14,80,23,77,20,74,17,83)(15,73,24,82,21,79,18,76)(25,89,34,86,31,95,28,92)(26,94,35,91,32,88,29,85)(27,87,36,96,33,93,30,90)(37,107,46,104,43,101,40,98)(38,100,47,97,44,106,41,103)(39,105,48,102,45,99,42,108)(49,109,58,118,55,115,52,112)(50,114,59,111,56,120,53,117)(51,119,60,116,57,113,54,110) );
G=PermutationGroup([[(1,49,37,31,22),(2,50,38,32,23),(3,51,39,33,24),(4,52,40,34,13),(5,53,41,35,14),(6,54,42,36,15),(7,55,43,25,16),(8,56,44,26,17),(9,57,45,27,18),(10,58,46,28,19),(11,59,47,29,20),(12,60,48,30,21),(61,112,98,86,75),(62,113,99,87,76),(63,114,100,88,77),(64,115,101,89,78),(65,116,102,90,79),(66,117,103,91,80),(67,118,104,92,81),(68,119,105,93,82),(69,120,106,94,83),(70,109,107,95,84),(71,110,108,96,73),(72,111,97,85,74)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,46,43,40),(38,47,44,41),(39,48,45,42),(49,58,55,52),(50,59,56,53),(51,60,57,54),(61,64,67,70),(62,65,68,71),(63,66,69,72),(73,76,79,82),(74,77,80,83),(75,78,81,84),(85,88,91,94),(86,89,92,95),(87,90,93,96),(97,100,103,106),(98,101,104,107),(99,102,105,108),(109,112,115,118),(110,113,116,119),(111,114,117,120)], [(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)], [(1,70,10,67,7,64,4,61),(2,63,11,72,8,69,5,66),(3,68,12,65,9,62,6,71),(13,75,22,84,19,81,16,78),(14,80,23,77,20,74,17,83),(15,73,24,82,21,79,18,76),(25,89,34,86,31,95,28,92),(26,94,35,91,32,88,29,85),(27,87,36,96,33,93,30,90),(37,107,46,104,43,101,40,98),(38,100,47,97,44,106,41,103),(39,105,48,102,45,99,42,108),(49,109,58,118,55,115,52,112),(50,114,59,111,56,120,53,117),(51,119,60,116,57,113,54,110)]])
C5×C4.Dic3 is a maximal subgroup of
C60.28D4 C60.29D4 C12.6D20 C60.31D4 C60.96D4 D60⋊16C4 C60.105D4 C60.D4 D20.2Dic3 D60.5C4 D15⋊4M4(2) D20⋊19D6 D60⋊30C22 C60.63D4 C12.D20 C5×S3×M4(2)
90 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | S3 | Dic3 | D6 | Dic3 | M4(2) | C5×S3 | C4.Dic3 | C5×Dic3 | S3×C10 | C5×Dic3 | C5×M4(2) | C5×C4.Dic3 |
| kernel | C5×C4.Dic3 | C5×C3⋊C8 | C2×C60 | C60 | C2×C30 | C4.Dic3 | C3⋊C8 | C2×C12 | C12 | C2×C6 | C2×C20 | C20 | C20 | C2×C10 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 16 |
Matrix representation of C5×C4.Dic3 ►in GL2(𝔽241) generated by
| 205 | 0 |
| 0 | 205 |
| 177 | 0 |
| 0 | 64 |
| 60 | 0 |
| 0 | 4 |
| 0 | 1 |
| 177 | 0 |
G:=sub<GL(2,GF(241))| [205,0,0,205],[177,0,0,64],[60,0,0,4],[0,177,1,0] >;
C5×C4.Dic3 in GAP, Magma, Sage, TeX
C_5\times C_4.{\rm Dic}_3 % in TeX
G:=Group("C5xC4.Dic3"); // GroupNames label
G:=SmallGroup(240,55);
// by ID
G=gap.SmallGroup(240,55);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,120,505,69,5765]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations
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