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## G = S3×C4.Dic5order 480 = 25·3·5

### Direct product of S3 and C4.Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — S3×C4.Dic5
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — S3×C5⋊2C8 — S3×C4.Dic5
 Lower central C15 — C30 — S3×C4.Dic5
 Upper central C1 — C4 — C2×C4

Generators and relations for S3×C4.Dic5
G = < a,b,c,d,e | a3=b2=c4=1, d10=c2, e2=d5, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d9 >

Subgroups: 412 in 136 conjugacy classes, 64 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, S3, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C5×S3, C30, C30, C2×M4(2), C52C8, C52C8, C2×C20, C2×C20, C22×C10, S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C52C8, C4.Dic5, C4.Dic5, C22×C20, S3×M4(2), C3×C52C8, C153C8, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C2×C4.Dic5, S3×C52C8, D6.Dic5, C3×C4.Dic5, C60.7C4, S3×C2×C20, S3×C4.Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, M4(2), C22×C4, Dic5, D10, C4×S3, C22×S3, C2×M4(2), C2×Dic5, C22×D5, S3×C2×C4, S3×D5, C4.Dic5, C22×Dic5, S3×M4(2), S3×Dic5, C2×S3×D5, C2×C4.Dic5, C2×S3×Dic5, S3×C4.Dic5

Smallest permutation representation of S3×C4.Dic5
On 120 points
Generators in S120
(1 103 41)(2 104 42)(3 105 43)(4 106 44)(5 107 45)(6 108 46)(7 109 47)(8 110 48)(9 111 49)(10 112 50)(11 113 51)(12 114 52)(13 115 53)(14 116 54)(15 117 55)(16 118 56)(17 119 57)(18 120 58)(19 101 59)(20 102 60)(21 62 90)(22 63 91)(23 64 92)(24 65 93)(25 66 94)(26 67 95)(27 68 96)(28 69 97)(29 70 98)(30 71 99)(31 72 100)(32 73 81)(33 74 82)(34 75 83)(35 76 84)(36 77 85)(37 78 86)(38 79 87)(39 80 88)(40 61 89)
(21 90)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 81)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 89)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 101)(60 102)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)(81 96 91 86)(82 97 92 87)(83 98 93 88)(84 99 94 89)(85 100 95 90)(101 106 111 116)(102 107 112 117)(103 108 113 118)(104 109 114 119)(105 110 115 120)
(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 74 6 79 11 64 16 69)(2 63 7 68 12 73 17 78)(3 72 8 77 13 62 18 67)(4 61 9 66 14 71 19 76)(5 70 10 75 15 80 20 65)(21 58 26 43 31 48 36 53)(22 47 27 52 32 57 37 42)(23 56 28 41 33 46 38 51)(24 45 29 50 34 55 39 60)(25 54 30 59 35 44 40 49)(81 119 86 104 91 109 96 114)(82 108 87 113 92 118 97 103)(83 117 88 102 93 107 98 112)(84 106 89 111 94 116 99 101)(85 115 90 120 95 105 100 110)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G ··· 10N 12A 12B 12C 15A 15B 20A ··· 20H 20I ··· 20P 24A 24B 24C 24D 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 5 5 6 6 8 8 8 8 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 15 15 20 ··· 20 20 ··· 20 24 24 24 24 30 ··· 30 60 ··· 60 size 1 1 2 3 3 6 2 1 1 2 3 3 6 2 2 2 4 10 10 10 10 30 30 30 30 2 ··· 2 6 ··· 6 2 2 4 4 4 2 ··· 2 6 ··· 6 20 20 20 20 4 ··· 4 4 ··· 4

78 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 D6 D6 M4(2) Dic5 D10 Dic5 D10 Dic5 C4×S3 C4×S3 C4.Dic5 S3×D5 S3×M4(2) S3×Dic5 C2×S3×D5 S3×Dic5 S3×C4.Dic5 kernel S3×C4.Dic5 S3×C5⋊2C8 D6.Dic5 C3×C4.Dic5 C60.7C4 S3×C2×C20 S3×C20 C10×Dic3 S3×C2×C10 C4.Dic5 S3×C2×C4 C5⋊2C8 C2×C20 C5×S3 C4×S3 C4×S3 C2×Dic3 C2×C12 C22×S3 C20 C2×C10 S3 C2×C4 C5 C4 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 1 2 2 1 4 4 4 2 2 2 2 2 16 2 2 2 2 2 8

Matrix representation of S3×C4.Dic5 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 1 213 0 0 112 239
,
 240 0 0 0 0 240 0 0 0 0 240 28 0 0 0 1
,
 64 0 0 0 0 177 0 0 0 0 1 0 0 0 0 1
,
 6 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 64 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,1,112,0,0,213,239],[240,0,0,0,0,240,0,0,0,0,240,0,0,0,28,1],[64,0,0,0,0,177,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,64,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

S3×C4.Dic5 in GAP, Magma, Sage, TeX

S_3\times C_4.{\rm Dic}_5
% in TeX

G:=Group("S3xC4.Dic5");
// GroupNames label

G:=SmallGroup(480,363);
// by ID

G=gap.SmallGroup(480,363);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=1,d^10=c^2,e^2=d^5,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^9>;
// generators/relations

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