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## G = D5×C4.Dic3order 480 = 25·3·5

### Direct product of D5 and C4.Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D5×C4.Dic3
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×C3⋊C8 — D5×C4.Dic3
 Lower central C15 — C30 — D5×C4.Dic3
 Upper central C1 — C4 — C2×C4

Generators and relations for D5×C4.Dic3
G = < a,b,c,d,e | a5=b2=c4=1, d6=c2, e2=c2d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >

Subgroups: 476 in 136 conjugacy classes, 64 normal (50 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×4], C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C3⋊C8 [×2], C3⋊C8 [×2], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×2], C3×D5, C30, C30, C2×M4(2), C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8 [×2], C4.Dic3, C4.Dic3 [×3], C22×C12, C3×Dic5 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5, C5×M4(2), C2×C4×D5, C2×C4.Dic3, C5×C3⋊C8 [×2], C153C8 [×2], D5×C12 [×4], C6×Dic5, C2×C60, D5×C2×C6, D5×M4(2), D5×C3⋊C8 [×2], C20.32D6 [×2], C5×C4.Dic3, C60.7C4, D5×C2×C12, D5×C4.Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, Dic3 [×4], D6 [×3], M4(2) [×2], C22×C4, D10 [×3], C2×Dic3 [×6], C22×S3, C2×M4(2), C4×D5 [×2], C22×D5, C4.Dic3 [×2], C22×Dic3, S3×D5, C2×C4×D5, C2×C4.Dic3, D5×Dic3 [×2], C2×S3×D5, D5×M4(2), C2×D5×Dic3, D5×C4.Dic3

Smallest permutation representation of D5×C4.Dic3
On 120 points
Generators in S120
(1 73 35 21 70)(2 74 36 22 71)(3 75 25 23 72)(4 76 26 24 61)(5 77 27 13 62)(6 78 28 14 63)(7 79 29 15 64)(8 80 30 16 65)(9 81 31 17 66)(10 82 32 18 67)(11 83 33 19 68)(12 84 34 20 69)(37 96 59 116 102)(38 85 60 117 103)(39 86 49 118 104)(40 87 50 119 105)(41 88 51 120 106)(42 89 52 109 107)(43 90 53 110 108)(44 91 54 111 97)(45 92 55 112 98)(46 93 56 113 99)(47 94 57 114 100)(48 95 58 115 101)
(1 70)(2 71)(3 72)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 73)(22 74)(23 75)(24 76)(37 96)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 104)(50 105)(51 106)(52 107)(53 108)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)
(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 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 70 67 64)(62 71 68 65)(63 72 69 66)(73 82 79 76)(74 83 80 77)(75 84 81 78)(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 91 10 88 7 85 4 94)(2 96 11 93 8 90 5 87)(3 89 12 86 9 95 6 92)(13 105 22 102 19 99 16 108)(14 98 23 107 20 104 17 101)(15 103 24 100 21 97 18 106)(25 109 34 118 31 115 28 112)(26 114 35 111 32 120 29 117)(27 119 36 116 33 113 30 110)(37 68 46 65 43 62 40 71)(38 61 47 70 44 67 41 64)(39 66 48 63 45 72 42 69)(49 81 58 78 55 75 52 84)(50 74 59 83 56 80 53 77)(51 79 60 76 57 73 54 82)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 20E 20F 30A ··· 30F 40A ··· 40H 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 10 10 10 10 12 12 12 12 12 12 12 12 15 15 20 20 20 20 20 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 5 5 10 2 1 1 2 5 5 10 2 2 2 2 2 10 10 10 10 6 6 6 6 30 30 30 30 2 2 4 4 2 2 2 2 10 10 10 10 4 4 2 2 2 2 4 4 4 ··· 4 12 ··· 12 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + - + - + - + + + - + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D5 Dic3 D6 Dic3 D6 Dic3 M4(2) D10 D10 C4×D5 C4×D5 C4.Dic3 S3×D5 D5×Dic3 C2×S3×D5 D5×Dic3 D5×M4(2) D5×C4.Dic3 kernel D5×C4.Dic3 D5×C3⋊C8 C20.32D6 C5×C4.Dic3 C60.7C4 D5×C2×C12 D5×C12 C6×Dic5 D5×C2×C6 C2×C4×D5 C4.Dic3 C4×D5 C4×D5 C2×Dic5 C2×C20 C22×D5 C3×D5 C3⋊C8 C2×C12 C12 C2×C6 D5 C2×C4 C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 4 2 2 1 2 2 2 1 1 1 4 4 2 4 4 8 2 2 2 2 4 8

Matrix representation of D5×C4.Dic3 in GL4(𝔽241) generated by

 0 1 0 0 240 51 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 0 240 0 0 0 0 177 0 0 0 118 64
,
 240 0 0 0 0 240 0 0 0 0 60 0 0 0 99 4
,
 177 0 0 0 0 177 0 0 0 0 177 192 0 0 139 64
G:=sub<GL(4,GF(241))| [0,240,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,177,118,0,0,0,64],[240,0,0,0,0,240,0,0,0,0,60,99,0,0,0,4],[177,0,0,0,0,177,0,0,0,0,177,139,0,0,192,64] >;

D5×C4.Dic3 in GAP, Magma, Sage, TeX

D_5\times C_4.{\rm Dic}_3
% in TeX

G:=Group("D5xC4.Dic3");
// GroupNames label

G:=SmallGroup(480,358);
// by ID

G=gap.SmallGroup(480,358);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,219,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=1,d^6=c^2,e^2=c^2*d^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;
// generators/relations

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