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## G = C5×S3×D8order 480 = 25·3·5

### Direct product of C5, S3 and D8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×S3×D8
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×S3×D4 — C5×S3×D8
 Lower central C3 — C6 — C12 — C5×S3×D8
 Upper central C1 — C10 — C20 — C5×D8

Generators and relations for C5×S3×D8
G = < a,b,c,d,e | a5=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 484 in 152 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C8, C2×C4, D4, D4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, D8, D8, C2×D4, C20, C20, C2×C10, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C5×S3, C5×S3, C30, C30, C2×D8, C40, C40, C2×C20, C5×D4, C5×D4, C22×C10, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C40, C5×D8, C5×D8, D4×C10, S3×D8, C5×C3⋊C8, C120, S3×C20, C5×D12, C5×C3⋊D4, D4×C15, S3×C2×C10, C10×D8, S3×C40, C5×D24, C5×D4⋊S3, C15×D8, C5×S3×D4, C5×S3×D8
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, D8, C2×D4, C2×C10, C22×S3, C5×S3, C2×D8, C5×D4, C22×C10, S3×D4, S3×C10, C5×D8, D4×C10, S3×D8, S3×C2×C10, C10×D8, C5×S3×D4, C5×S3×D8

Smallest permutation representation of C5×S3×D8
On 120 points
Generators in S120
(1 62 90 52 74)(2 63 91 53 75)(3 64 92 54 76)(4 57 93 55 77)(5 58 94 56 78)(6 59 95 49 79)(7 60 96 50 80)(8 61 89 51 73)(9 29 110 22 45)(10 30 111 23 46)(11 31 112 24 47)(12 32 105 17 48)(13 25 106 18 41)(14 26 107 19 42)(15 27 108 20 43)(16 28 109 21 44)(33 119 82 103 66)(34 120 83 104 67)(35 113 84 97 68)(36 114 85 98 69)(37 115 86 99 70)(38 116 87 100 71)(39 117 88 101 72)(40 118 81 102 65)
(1 22 99)(2 23 100)(3 24 101)(4 17 102)(5 18 103)(6 19 104)(7 20 97)(8 21 98)(9 37 90)(10 38 91)(11 39 92)(12 40 93)(13 33 94)(14 34 95)(15 35 96)(16 36 89)(25 119 56)(26 120 49)(27 113 50)(28 114 51)(29 115 52)(30 116 53)(31 117 54)(32 118 55)(41 66 58)(42 67 59)(43 68 60)(44 69 61)(45 70 62)(46 71 63)(47 72 64)(48 65 57)(73 109 85)(74 110 86)(75 111 87)(76 112 88)(77 105 81)(78 106 82)(79 107 83)(80 108 84)
(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(25 119)(26 120)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 65)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)
(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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 16)(14 15)(17 22)(18 21)(19 20)(23 24)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)(37 40)(38 39)(41 44)(42 43)(45 48)(46 47)(49 50)(51 56)(52 55)(53 54)(57 62)(58 61)(59 60)(63 64)(65 70)(66 69)(67 68)(71 72)(73 78)(74 77)(75 76)(79 80)(81 86)(82 85)(83 84)(87 88)(89 94)(90 93)(91 92)(95 96)(97 104)(98 103)(99 102)(100 101)(105 110)(106 109)(107 108)(111 112)(113 120)(114 119)(115 118)(116 117)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 10A 10B 10C 10D 10E ··· 10L 10M ··· 10T 10U ··· 10AB 12 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 24A 24B 30A 30B 30C 30D 30E ··· 30L 40A ··· 40H 40I ··· 40P 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 2 2 2 3 4 4 5 5 5 5 6 6 6 8 8 8 8 10 10 10 10 10 ··· 10 10 ··· 10 10 ··· 10 12 15 15 15 15 20 20 20 20 20 20 20 20 24 24 30 30 30 30 30 ··· 30 40 ··· 40 40 ··· 40 60 60 60 60 120 ··· 120 size 1 1 3 3 4 4 12 12 2 2 6 1 1 1 1 2 8 8 2 2 6 6 1 1 1 1 3 ··· 3 4 ··· 4 12 ··· 12 4 2 2 2 2 2 2 2 2 6 6 6 6 4 4 2 2 2 2 8 ··· 8 2 ··· 2 6 ··· 6 4 4 4 4 4 ··· 4

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 D8 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C5×D8 S3×D4 S3×D8 C5×S3×D4 C5×S3×D8 kernel C5×S3×D8 S3×C40 C5×D24 C5×D4⋊S3 C15×D8 C5×S3×D4 S3×D8 S3×C8 D24 D4⋊S3 C3×D8 S3×D4 C5×D8 C5×Dic3 S3×C10 C40 C5×D4 C5×S3 D8 Dic3 D6 C8 D4 S3 C10 C5 C2 C1 # reps 1 1 1 2 1 2 4 4 4 8 4 8 1 1 1 1 2 4 4 4 4 4 8 16 1 2 4 8

Matrix representation of C5×S3×D8 in GL4(𝔽241) generated by

 98 0 0 0 0 98 0 0 0 0 91 0 0 0 0 91
,
 0 240 0 0 1 240 0 0 0 0 1 0 0 0 0 1
,
 1 240 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 240 0 0 0 0 240 0 0 0 0 230 11 0 0 230 230
,
 240 0 0 0 0 240 0 0 0 0 11 230 0 0 230 230
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,91,0,0,0,0,91],[0,1,0,0,240,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,240,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,230,230,0,0,11,230],[240,0,0,0,0,240,0,0,0,0,11,230,0,0,230,230] >;

C5×S3×D8 in GAP, Magma, Sage, TeX

C_5\times S_3\times D_8
% in TeX

G:=Group("C5xS3xD8");
// GroupNames label

G:=SmallGroup(480,789);
// by ID

G=gap.SmallGroup(480,789);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,471,2111,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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