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## G = D5×C8⋊S3order 480 = 25·3·5

### Direct product of D5 and C8⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D5×C8⋊S3
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — C4×S3×D5 — D5×C8⋊S3
 Lower central C15 — C30 — D5×C8⋊S3
 Upper central C1 — C4 — C8

Generators and relations for D5×C8⋊S3
G = < a,b,c,d,e | a5=b2=c8=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c5, ede=d-1 >

Subgroups: 636 in 136 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×3], C22 [×5], C5, S3 [×2], C6, C6 [×2], C8, C8 [×3], C2×C4 [×6], C23, D5 [×2], D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6 [×3], C2×C6, C15, C2×C8 [×2], M4(2) [×4], C22×C4, Dic5, Dic5, C20, C20, D10, D10 [×3], C2×C10, C3⋊C8, C3⋊C8, C24, C24, C4×S3, C4×S3 [×3], C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5 [×2], D15, C30, C2×M4(2), C52C8, C52C8, C40, C40, C4×D5, C4×D5 [×3], C2×Dic5, C2×C20, C22×D5, C8⋊S3, C8⋊S3 [×3], C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C8×D5, C8×D5, C8⋊D5 [×2], C4.Dic5, C5×M4(2), C2×C4×D5, C2×C8⋊S3, C5×C3⋊C8, C3×C52C8, C153C8, C120, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, D5×M4(2), D5×C3⋊C8, D6.Dic5, D30.5C4, D5×C24, C5×C8⋊S3, C40⋊S3, C4×S3×D5, D5×C8⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], M4(2) [×2], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C2×M4(2), C4×D5 [×2], C22×D5, C8⋊S3 [×2], S3×C2×C4, S3×D5, C2×C4×D5, C2×C8⋊S3, C2×S3×D5, D5×M4(2), C4×S3×D5, D5×C8⋊S3

Smallest permutation representation of D5×C8⋊S3
On 120 points
Generators in S120
(1 73 113 65 99)(2 74 114 66 100)(3 75 115 67 101)(4 76 116 68 102)(5 77 117 69 103)(6 78 118 70 104)(7 79 119 71 97)(8 80 120 72 98)(9 96 54 109 45)(10 89 55 110 46)(11 90 56 111 47)(12 91 49 112 48)(13 92 50 105 41)(14 93 51 106 42)(15 94 52 107 43)(16 95 53 108 44)(17 34 29 63 83)(18 35 30 64 84)(19 36 31 57 85)(20 37 32 58 86)(21 38 25 59 87)(22 39 26 60 88)(23 40 27 61 81)(24 33 28 62 82)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 109)(10 110)(11 111)(12 112)(13 105)(14 106)(15 107)(16 108)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 82)(34 83)(35 84)(36 85)(37 86)(38 87)(39 88)(40 81)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 89)(56 90)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)
(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 50 64)(2 51 57)(3 52 58)(4 53 59)(5 54 60)(6 55 61)(7 56 62)(8 49 63)(9 39 69)(10 40 70)(11 33 71)(12 34 72)(13 35 65)(14 36 66)(15 37 67)(16 38 68)(17 120 48)(18 113 41)(19 114 42)(20 115 43)(21 116 44)(22 117 45)(23 118 46)(24 119 47)(25 102 95)(26 103 96)(27 104 89)(28 97 90)(29 98 91)(30 99 92)(31 100 93)(32 101 94)(73 105 84)(74 106 85)(75 107 86)(76 108 87)(77 109 88)(78 110 81)(79 111 82)(80 112 83)
(2 6)(4 8)(9 39)(10 36)(11 33)(12 38)(13 35)(14 40)(15 37)(16 34)(17 44)(18 41)(19 46)(20 43)(21 48)(22 45)(23 42)(24 47)(25 91)(26 96)(27 93)(28 90)(29 95)(30 92)(31 89)(32 94)(49 59)(50 64)(51 61)(52 58)(53 63)(54 60)(55 57)(56 62)(66 70)(68 72)(74 78)(76 80)(81 106)(82 111)(83 108)(84 105)(85 110)(86 107)(87 112)(88 109)(98 102)(100 104)(114 118)(116 120)

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

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

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

72 conjugacy classes

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

72 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 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D5 D6 D6 D6 M4(2) D10 D10 D10 C4×S3 C4×S3 C4×D5 C4×D5 C8⋊S3 S3×D5 C2×S3×D5 D5×M4(2) C4×S3×D5 D5×C8⋊S3 kernel D5×C8⋊S3 D5×C3⋊C8 D6.Dic5 D30.5C4 D5×C24 C5×C8⋊S3 C40⋊S3 C4×S3×D5 D5×Dic3 S3×Dic5 D30.C2 C2×S3×D5 C8×D5 C8⋊S3 C5⋊2C8 C40 C4×D5 C3×D5 C3⋊C8 C24 C4×S3 Dic5 D10 Dic3 D6 D5 C8 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 1 2 1 1 1 4 2 2 2 2 2 4 4 8 2 2 4 4 8

Matrix representation of D5×C8⋊S3 in GL6(𝔽241)

 51 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 51 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 239 0 0 0 0 88 177
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 240 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 240 240 0 0 0 0 0 0 1 0 0 0 0 0 64 240

G:=sub<GL(6,GF(241))| [51,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,51,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,88,0,0,0,0,239,177],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,1,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,0,1,64,0,0,0,0,0,240] >;

D5×C8⋊S3 in GAP, Magma, Sage, TeX

D_5\times C_8\rtimes S_3
% in TeX

G:=Group("D5xC8:S3");
// GroupNames label

G:=SmallGroup(480,320);
// by ID

G=gap.SmallGroup(480,320);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,58,80,1356,18822]);
// Polycyclic

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

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