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## G = D5×Q8⋊2S3order 480 = 25·3·5

### Direct product of D5 and Q8⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D5×Q8⋊2S3
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×D12 — D5×Q8⋊2S3
 Lower central C15 — C30 — C60 — D5×Q8⋊2S3
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D5×Q82S3
G = < a,b,c,d,e,f | a5=b2=c4=e3=f2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fcf=c-1, ce=ec, de=ed, fdf=c-1d, fef=e-1 >

Subgroups: 876 in 136 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C23, D5, D5, C10, C10, C12, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, D12, D12, C2×C12, C3×Q8, C3×Q8, C22×S3, C5×S3, C3×D5, D15, C30, C2×SD16, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C2×C3⋊C8, Q82S3, Q82S3, C2×D12, C6×Q8, C3×Dic5, C3×Dic5, C60, C60, S3×D5, C6×D5, S3×C10, D30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, C2×Q82S3, C5×C3⋊C8, C153C8, C5⋊D12, C3×Dic10, C3×Dic10, D5×C12, D5×C12, C5×D12, D60, Q8×C15, C2×S3×D5, D5×SD16, D5×C3⋊C8, C20.D6, C15⋊SD16, C5×Q82S3, Q82D15, D5×D12, C3×Q8×D5, D5×Q82S3
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C3⋊D4, C22×S3, C2×SD16, C22×D5, Q82S3, C2×C3⋊D4, S3×D5, D4×D5, C2×Q82S3, C2×S3×D5, D5×SD16, D5×C3⋊D4, D5×Q82S3

Smallest permutation representation of D5×Q82S3
On 120 points
Generators in S120
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)(82 85)(83 84)(87 90)(88 89)(92 95)(93 94)(97 100)(98 99)(102 105)(103 104)(107 110)(108 109)(112 115)(113 114)(117 120)(118 119)
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)(61 91 76 106)(62 92 77 107)(63 93 78 108)(64 94 79 109)(65 95 80 110)(66 96 81 111)(67 97 82 112)(68 98 83 113)(69 99 84 114)(70 100 85 115)(71 101 86 116)(72 102 87 117)(73 103 88 118)(74 104 89 119)(75 105 90 120)
(1 79 19 64)(2 80 20 65)(3 76 16 61)(4 77 17 62)(5 78 18 63)(6 81 21 66)(7 82 22 67)(8 83 23 68)(9 84 24 69)(10 85 25 70)(11 86 26 71)(12 87 27 72)(13 88 28 73)(14 89 29 74)(15 90 30 75)(31 106 46 91)(32 107 47 92)(33 108 48 93)(34 109 49 94)(35 110 50 95)(36 111 51 96)(37 112 52 97)(38 113 53 98)(39 114 54 99)(40 115 55 100)(41 116 56 101)(42 117 57 102)(43 118 58 103)(44 119 59 104)(45 120 60 105)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(91 101 96)(92 102 97)(93 103 98)(94 104 99)(95 105 100)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 56)(37 57)(38 58)(39 59)(40 60)(41 51)(42 52)(43 53)(44 54)(45 55)(61 91)(62 92)(63 93)(64 94)(65 95)(66 101)(67 102)(68 103)(69 104)(70 105)(71 96)(72 97)(73 98)(74 99)(75 100)(76 106)(77 107)(78 108)(79 109)(80 110)(81 116)(82 117)(83 118)(84 119)(85 120)(86 111)(87 112)(88 113)(89 114)(90 115)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 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 8 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 SD16 D10 D10 D10 C3⋊D4 C3⋊D4 Q8⋊2S3 S3×D5 D4×D5 C2×S3×D5 D5×SD16 D5×C3⋊D4 D5×Q8⋊2S3 kernel D5×Q8⋊2S3 D5×C3⋊C8 C20.D6 C15⋊SD16 C5×Q8⋊2S3 Q8⋊2D15 D5×D12 C3×Q8×D5 Q8×D5 C3×Dic5 C6×D5 Q8⋊2S3 Dic10 C4×D5 C5×Q8 C3×D5 C3⋊C8 D12 C3×Q8 Dic5 D10 D5 Q8 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 2 2 2 2 2 2 2 2 4 4 2

Matrix representation of D5×Q82S3 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 189 1 0 0 0 0 240 0 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 1 189 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 103 5 0 0 0 0 47 138
,
 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 0 193 146 0 0 0 0 75 48
,
 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 7 240

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,189,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,189,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,47,0,0,0,0,5,138],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,193,75,0,0,0,0,146,48],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,7,0,0,0,0,0,240] >;

D5×Q82S3 in GAP, Magma, Sage, TeX

D_5\times Q_8\rtimes_2S_3
% in TeX

G:=Group("D5xQ8:2S3");
// GroupNames label

G:=SmallGroup(480,577);
// by ID

G=gap.SmallGroup(480,577);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,135,100,346,185,80,1356,18822]);
// Polycyclic

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

׿
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