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## G = C5×C4.Dic3order 240 = 24·3·5

### Direct product of C5 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×C4.Dic3
 Chief series C1 — C3 — C6 — C12 — C60 — C5×C3⋊C8 — C5×C4.Dic3
 Lower central C3 — C6 — C5×C4.Dic3
 Upper central C1 — C20 — C2×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 >

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 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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