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## G = S3×C8⋊D5order 480 = 25·3·5

### Direct product of S3 and C8⋊D5

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C8⋊D5
G = < a,b,c,d,e | a3=b2=c8=d5=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], S3, C6, C6, C8, C8 [×3], C2×C4 [×6], C23, D5 [×2], C10, C10 [×2], 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 [×2], C3×D5, D15, C30, C2×M4(2), C52C8, C52C8, C40, C40, C4×D5, C4×D5 [×3], C2×Dic5, C2×C20, C22×D5, S3×C8, S3×C8, C8⋊S3 [×2], C4.Dic3, C3×M4(2), S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C8⋊D5, C8⋊D5 [×3], C2×C52C8, C2×C40, C2×C4×D5, S3×M4(2), C5×C3⋊C8, C3×C52C8, C153C8, C120, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, C2×C8⋊D5, S3×C52C8, C20.32D6, D30.5C4, C3×C8⋊D5, S3×C40, C40⋊S3, C4×S3×D5, S3×C8⋊D5
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, S3×C2×C4, S3×D5, C8⋊D5 [×2], C2×C4×D5, S3×M4(2), C2×S3×D5, C2×C8⋊D5, C4×S3×D5, S3×C8⋊D5

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

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

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

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

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

78 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⋊D5 S3×D5 S3×M4(2) C2×S3×D5 C4×S3×D5 S3×C8⋊D5 kernel S3×C8⋊D5 S3×C5⋊2C8 C20.32D6 D30.5C4 C3×C8⋊D5 S3×C40 C40⋊S3 C4×S3×D5 D5×Dic3 S3×Dic5 D30.C2 C2×S3×D5 C8⋊D5 S3×C8 C5⋊2C8 C40 C4×D5 C5×S3 C3⋊C8 C24 C4×S3 Dic5 D10 Dic3 D6 S3 C8 C5 C4 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 16 2 2 2 4 8

Matrix representation of S3×C8⋊D5 in GL4(𝔽241) generated by

 240 240 0 0 1 0 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 0 216 101 0 0 140 25
,
 1 0 0 0 0 1 0 0 0 0 189 240 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 189 240 0 0 52 52
G:=sub<GL(4,GF(241))| [240,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,216,140,0,0,101,25],[1,0,0,0,0,1,0,0,0,0,189,1,0,0,240,0],[1,0,0,0,0,1,0,0,0,0,189,52,0,0,240,52] >;

S3×C8⋊D5 in GAP, Magma, Sage, TeX

S_3\times C_8\rtimes D_5
% in TeX

G:=Group("S3xC8:D5");
// GroupNames label

G:=SmallGroup(480,321);
// by ID

G=gap.SmallGroup(480,321);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,58,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^8=d^5=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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