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G = C5×C4.Dic3order 240 = 24·3·5

Direct product of C5 and C4.Dic3

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C4.Dic3, C60.11C4, C12.1C20, C20.59D6, C1515M4(2), C20.7Dic3, C60.76C22, C3⋊C85C10, C4.(C5×Dic3), (C2×C20).8S3, C6.6(C2×C20), (C2×C6).3C20, C32(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

C1C6 — C5×C4.Dic3
C1C3C6C12C60C5×C3⋊C8 — C5×C4.Dic3
C3C6 — C5×C4.Dic3
C1C20C2×C20

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 >

2C2
2C6
2C10
3C8
3C8
2C30
3M4(2)
3C40
3C40
3C5×M4(2)

Smallest permutation representation of C5×C4.Dic3
On 120 points
Generators in S120
(1 58 40 31 13)(2 59 41 32 14)(3 60 42 33 15)(4 49 43 34 16)(5 50 44 35 17)(6 51 45 36 18)(7 52 46 25 19)(8 53 47 26 20)(9 54 48 27 21)(10 55 37 28 22)(11 56 38 29 23)(12 57 39 30 24)(61 115 99 87 82)(62 116 100 88 83)(63 117 101 89 84)(64 118 102 90 73)(65 119 103 91 74)(66 120 104 92 75)(67 109 105 93 76)(68 110 106 94 77)(69 111 107 95 78)(70 112 108 96 79)(71 113 97 85 80)(72 114 98 86 81)
(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 61 10 70 7 67 4 64)(2 66 11 63 8 72 5 69)(3 71 12 68 9 65 6 62)(13 82 22 79 19 76 16 73)(14 75 23 84 20 81 17 78)(15 80 24 77 21 74 18 83)(25 93 34 90 31 87 28 96)(26 86 35 95 32 92 29 89)(27 91 36 88 33 85 30 94)(37 108 46 105 43 102 40 99)(38 101 47 98 44 107 41 104)(39 106 48 103 45 100 42 97)(49 118 58 115 55 112 52 109)(50 111 59 120 56 117 53 114)(51 116 60 113 57 110 54 119)

G:=sub<Sym(120)| (1,58,40,31,13)(2,59,41,32,14)(3,60,42,33,15)(4,49,43,34,16)(5,50,44,35,17)(6,51,45,36,18)(7,52,46,25,19)(8,53,47,26,20)(9,54,48,27,21)(10,55,37,28,22)(11,56,38,29,23)(12,57,39,30,24)(61,115,99,87,82)(62,116,100,88,83)(63,117,101,89,84)(64,118,102,90,73)(65,119,103,91,74)(66,120,104,92,75)(67,109,105,93,76)(68,110,106,94,77)(69,111,107,95,78)(70,112,108,96,79)(71,113,97,85,80)(72,114,98,86,81), (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), (1,61,10,70,7,67,4,64)(2,66,11,63,8,72,5,69)(3,71,12,68,9,65,6,62)(13,82,22,79,19,76,16,73)(14,75,23,84,20,81,17,78)(15,80,24,77,21,74,18,83)(25,93,34,90,31,87,28,96)(26,86,35,95,32,92,29,89)(27,91,36,88,33,85,30,94)(37,108,46,105,43,102,40,99)(38,101,47,98,44,107,41,104)(39,106,48,103,45,100,42,97)(49,118,58,115,55,112,52,109)(50,111,59,120,56,117,53,114)(51,116,60,113,57,110,54,119)>;

G:=Group( (1,58,40,31,13)(2,59,41,32,14)(3,60,42,33,15)(4,49,43,34,16)(5,50,44,35,17)(6,51,45,36,18)(7,52,46,25,19)(8,53,47,26,20)(9,54,48,27,21)(10,55,37,28,22)(11,56,38,29,23)(12,57,39,30,24)(61,115,99,87,82)(62,116,100,88,83)(63,117,101,89,84)(64,118,102,90,73)(65,119,103,91,74)(66,120,104,92,75)(67,109,105,93,76)(68,110,106,94,77)(69,111,107,95,78)(70,112,108,96,79)(71,113,97,85,80)(72,114,98,86,81), (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), (1,61,10,70,7,67,4,64)(2,66,11,63,8,72,5,69)(3,71,12,68,9,65,6,62)(13,82,22,79,19,76,16,73)(14,75,23,84,20,81,17,78)(15,80,24,77,21,74,18,83)(25,93,34,90,31,87,28,96)(26,86,35,95,32,92,29,89)(27,91,36,88,33,85,30,94)(37,108,46,105,43,102,40,99)(38,101,47,98,44,107,41,104)(39,106,48,103,45,100,42,97)(49,118,58,115,55,112,52,109)(50,111,59,120,56,117,53,114)(51,116,60,113,57,110,54,119) );

G=PermutationGroup([(1,58,40,31,13),(2,59,41,32,14),(3,60,42,33,15),(4,49,43,34,16),(5,50,44,35,17),(6,51,45,36,18),(7,52,46,25,19),(8,53,47,26,20),(9,54,48,27,21),(10,55,37,28,22),(11,56,38,29,23),(12,57,39,30,24),(61,115,99,87,82),(62,116,100,88,83),(63,117,101,89,84),(64,118,102,90,73),(65,119,103,91,74),(66,120,104,92,75),(67,109,105,93,76),(68,110,106,94,77),(69,111,107,95,78),(70,112,108,96,79),(71,113,97,85,80),(72,114,98,86,81)], [(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)], [(1,61,10,70,7,67,4,64),(2,66,11,63,8,72,5,69),(3,71,12,68,9,65,6,62),(13,82,22,79,19,76,16,73),(14,75,23,84,20,81,17,78),(15,80,24,77,21,74,18,83),(25,93,34,90,31,87,28,96),(26,86,35,95,32,92,29,89),(27,91,36,88,33,85,30,94),(37,108,46,105,43,102,40,99),(38,101,47,98,44,107,41,104),(39,106,48,103,45,100,42,97),(49,118,58,115,55,112,52,109),(50,111,59,120,56,117,53,114),(51,116,60,113,57,110,54,119)])

C5×C4.Dic3 is a maximal subgroup of
C60.28D4  C60.29D4  C12.6D20  C60.31D4  C60.96D4  D6016C4  C60.105D4  C60.D4  D20.2Dic3  D60.5C4  D154M4(2)  D2019D6  D6030C22  C60.63D4  C12.D20  C5×S3×M4(2)

90 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B5C5D6A6B6C8A8B8C8D10A10B10C10D10E10F10G10H12A12B12C12D15A15B15C15D20A···20H20I20J20K20L30A···30L40A···40P60A···60P
order1223444555566688881010101010101010121212121515151520···202020202030···3040···4060···60
size11221121111222666611112222222222221···122222···26···62···2

90 irreducible representations

dim1111111111222222222222
type++++-+-
imageC1C2C2C4C4C5C10C10C20C20S3Dic3D6Dic3M4(2)C5×S3C4.Dic3C5×Dic3S3×C10C5×Dic3C5×M4(2)C5×C4.Dic3
kernelC5×C4.Dic3C5×C3⋊C8C2×C60C60C2×C30C4.Dic3C3⋊C8C2×C12C12C2×C6C2×C20C20C20C2×C10C15C2×C4C5C4C4C22C3C1
# reps12122484881111244444816

Matrix representation of C5×C4.Dic3 in GL2(𝔽241) generated by

2050
0205
,
1770
064
,
600
04
,
01
1770
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

Export

Subgroup lattice of C5×C4.Dic3 in TeX

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