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## G = S3×D5⋊C8order 480 = 25·3·5

### Direct product of S3 and D5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — S3×D5⋊C8
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C3×C5⋊C8 — S3×C5⋊C8 — S3×D5⋊C8
 Lower central C15 — S3×D5⋊C8
 Upper central C1 — C4

Generators and relations for S3×D5⋊C8
G = < a,b,c,d,e | a3=b2=c5=d2=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ece-1=c3, ede-1=c2d >

Subgroups: 676 in 152 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C2×C4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C22×C8, C5⋊C8, C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, D5⋊C8, D5⋊C8, C2×C5⋊C8, C2×C4×D5, S3×C2×C8, C3×C5⋊C8, C15⋊C8, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, C2×D5⋊C8, S3×C5⋊C8, D15⋊C8, C3×D5⋊C8, C60.C4, C4×S3×D5, S3×D5⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, C22×C4, F5, C4×S3, C22×S3, C22×C8, C2×F5, S3×C8, S3×C2×C4, D5⋊C8, C22×F5, S3×C2×C8, S3×F5, C2×D5⋊C8, C2×S3×F5, S3×D5⋊C8

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 8 8 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8 S3 D6 D6 C4×S3 C4×S3 S3×C8 F5 C2×F5 C2×F5 C2×F5 D5⋊C8 S3×F5 C2×S3×F5 S3×D5⋊C8 kernel S3×D5⋊C8 S3×C5⋊C8 D15⋊C8 C3×D5⋊C8 C60.C4 C4×S3×D5 D5×Dic3 S3×C20 C4×D15 C2×S3×D5 S3×D5 D5⋊C8 C5⋊C8 C4×D5 C20 D10 D5 C4×S3 Dic3 C12 D6 S3 C4 C2 C1 # reps 1 2 2 1 1 1 2 2 2 2 16 1 2 1 2 2 8 1 1 1 1 4 1 1 2

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

 240 240 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 0 240 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 240 240 240 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 240 240 240 240
,
 177 0 0 0 0 0 0 177 0 0 0 0 0 0 0 171 2 171 0 0 72 0 70 70 0 0 70 70 0 72 0 0 171 2 171 0

G:=sub<GL(6,GF(241))| [240,1,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,240,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,240,0,0,0,1,0,240,0,0,1,0,0,240,0,0,0,0,0,240],[177,0,0,0,0,0,0,177,0,0,0,0,0,0,0,72,70,171,0,0,171,0,70,2,0,0,2,70,0,171,0,0,171,70,72,0] >;

S3×D5⋊C8 in GAP, Magma, Sage, TeX

S_3\times D_5\rtimes C_8
% in TeX

G:=Group("S3xD5:C8");
// GroupNames label

G:=SmallGroup(480,986);
// by ID

G=gap.SmallGroup(480,986);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,100,80,1356,9414,2379]);
// Polycyclic

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

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