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## G = C2×S3×C5⋊D4order 480 = 25·3·5

### Direct product of C2, S3 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×S3×C5⋊D4
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C22×S3×D5 — C2×S3×C5⋊D4
 Lower central C15 — C30 — C2×S3×C5⋊D4
 Upper central C1 — C22 — C23

Generators and relations for C2×S3×C5⋊D4
G = < a,b,c,d,e,f | a2=b3=c2=d5=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 2428 in 472 conjugacy classes, 132 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×4], C22, C22 [×2], C22 [×36], C5, S3 [×4], S3 [×4], C6, C6 [×2], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×4], C10, C10 [×2], C10 [×8], Dic3 [×2], C12 [×2], D6 [×8], D6 [×22], C2×C6, C2×C6 [×2], C2×C6 [×6], C15, C22×C4, C2×D4 [×12], C24 [×2], Dic5 [×2], Dic5 [×2], D10 [×2], D10 [×14], C2×C10, C2×C10 [×2], C2×C10 [×20], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3 [×2], C22×S3 [×4], C22×S3 [×13], C22×C6, C22×C6, C5×S3 [×4], C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C30 [×2], C30 [×2], C22×D4, C2×Dic5, C2×Dic5 [×5], C5⋊D4 [×4], C5⋊D4 [×12], C22×D5, C22×D5 [×9], C22×C10, C22×C10 [×10], S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23, S3×C23, C3×Dic5 [×2], Dic15 [×2], S3×D5 [×8], C6×D5 [×2], C6×D5 [×2], S3×C10 [×8], S3×C10 [×10], D30 [×2], D30 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C22×Dic5, C2×C5⋊D4, C2×C5⋊D4 [×11], C23×D5, C23×C10, C2×S3×D4, S3×Dic5 [×4], C15⋊D4 [×4], C5⋊D12 [×4], C6×Dic5, C3×C5⋊D4 [×4], C2×Dic15, C157D4 [×4], C2×S3×D5 [×4], C2×S3×D5 [×4], D5×C2×C6, S3×C2×C10 [×2], S3×C2×C10 [×4], S3×C2×C10 [×4], C22×D15, C22×C30, C22×C5⋊D4, C2×S3×Dic5, C2×C15⋊D4, C2×C5⋊D12, S3×C5⋊D4 [×8], C6×C5⋊D4, C2×C157D4, C22×S3×D5, S3×C22×C10, C2×S3×C5⋊D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C22×S3 [×7], C22×D4, C5⋊D4 [×4], C22×D5 [×7], S3×D4 [×2], S3×C23, S3×D5, C2×C5⋊D4 [×6], C23×D5, C2×S3×D4, C2×S3×D5 [×3], C22×C5⋊D4, S3×C5⋊D4 [×2], C22×S3×D5, C2×S3×C5⋊D4

Smallest permutation representation of C2×S3×C5⋊D4
On 120 points
Generators in S120
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(36 41)(37 42)(38 43)(39 44)(40 45)(51 56)(52 57)(53 58)(54 59)(55 60)(66 71)(67 72)(68 73)(69 74)(70 75)(81 86)(82 87)(83 88)(84 89)(85 90)(96 101)(97 102)(98 103)(99 104)(100 105)(111 116)(112 117)(113 118)(114 119)(115 120)
(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 91 16 106)(2 95 17 110)(3 94 18 109)(4 93 19 108)(5 92 20 107)(6 96 21 111)(7 100 22 115)(8 99 23 114)(9 98 24 113)(10 97 25 112)(11 101 26 116)(12 105 27 120)(13 104 28 119)(14 103 29 118)(15 102 30 117)(31 76 46 61)(32 80 47 65)(33 79 48 64)(34 78 49 63)(35 77 50 62)(36 81 51 66)(37 85 52 70)(38 84 53 69)(39 83 54 68)(40 82 55 67)(41 86 56 71)(42 90 57 75)(43 89 58 74)(44 88 59 73)(45 87 60 72)
(2 5)(3 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(31 46)(32 50)(33 49)(34 48)(35 47)(36 51)(37 55)(38 54)(39 53)(40 52)(41 56)(42 60)(43 59)(44 58)(45 57)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)(82 85)(83 84)(87 90)(88 89)(91 106)(92 110)(93 109)(94 108)(95 107)(96 111)(97 115)(98 114)(99 113)(100 112)(101 116)(102 120)(103 119)(104 118)(105 117)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10N 10O ··· 10AD 12A 12B 15A 15B 30A ··· 30N order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 15 15 30 ··· 30 size 1 1 1 1 2 2 3 3 3 3 6 6 10 10 30 30 2 10 10 30 30 2 2 2 2 2 4 4 20 20 2 ··· 2 6 ··· 6 20 20 4 4 4 ··· 4

78 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 D6 D10 D10 C5⋊D4 S3×D4 S3×D5 C2×S3×D5 S3×C5⋊D4 kernel C2×S3×C5⋊D4 C2×S3×Dic5 C2×C15⋊D4 C2×C5⋊D12 S3×C5⋊D4 C6×C5⋊D4 C2×C15⋊7D4 C22×S3×D5 S3×C22×C10 C2×C5⋊D4 S3×C10 S3×C23 C2×Dic5 C5⋊D4 C22×D5 C22×C10 C22×S3 C22×C6 D6 C10 C23 C22 C2 # reps 1 1 1 1 8 1 1 1 1 1 4 2 1 4 1 1 12 2 16 2 2 6 8

Matrix representation of C2×S3×C5⋊D4 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 1 0 0 0 0 60 60 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 60 60
,
 17 60 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 14 45 0 0 39 47 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 17 60 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,60,1,0,0,60,0],[1,0,0,0,0,1,0,0,0,0,1,60,0,0,0,60],[17,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[14,39,0,0,45,47,0,0,0,0,60,0,0,0,0,60],[1,17,0,0,0,60,0,0,0,0,1,0,0,0,0,1] >;

C2×S3×C5⋊D4 in GAP, Magma, Sage, TeX

C_2\times S_3\times C_5\rtimes D_4
% in TeX

G:=Group("C2xS3xC5:D4");
// GroupNames label

G:=SmallGroup(480,1123);
// by ID

G=gap.SmallGroup(480,1123);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,1356,18822]);
// Polycyclic

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

׿
×
𝔽