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## G = C5×S3×C22⋊C4order 480 = 25·3·5

### Direct product of C5, S3 and C22⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×S3×C22⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — S3×C22×C10 — C5×S3×C22⋊C4
 Lower central C3 — C6 — C5×S3×C22⋊C4
 Upper central C1 — C2×C10 — C5×C22⋊C4

Generators and relations for C5×S3×C22⋊C4
G = < a,b,c,d,e,f | a5=b3=c2=d2=e2=f4=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, fdf-1=de=ed, ef=fe >

Subgroups: 660 in 264 conjugacy classes, 98 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, S3, C6, C6, C6, C2×C4, C2×C4, C23, C23, C10, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C22×C4, C24, C20, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×S3, C22×C6, C5×S3, C5×S3, C30, C30, C30, C2×C22⋊C4, C2×C20, C2×C20, C22×C10, C22×C10, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C23, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C30, C2×C30, C5×C22⋊C4, C5×C22⋊C4, C22×C20, C23×C10, S3×C22⋊C4, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, S3×C2×C10, S3×C2×C10, C22×C30, C10×C22⋊C4, C5×D6⋊C4, C5×C6.D4, C15×C22⋊C4, S3×C2×C20, S3×C22×C10, C5×S3×C22⋊C4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C23, C10, D6, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, S3×C2×C4, S3×D4, S3×C10, C5×C22⋊C4, C22×C20, D4×C10, S3×C22⋊C4, S3×C20, S3×C2×C10, C10×C22⋊C4, S3×C2×C20, C5×S3×D4, C5×S3×C22⋊C4

Smallest permutation representation of C5×S3×C22⋊C4
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 61 20)(2 62 16)(3 63 17)(4 64 18)(5 65 19)(6 42 60)(7 43 56)(8 44 57)(9 45 58)(10 41 59)(11 84 22)(12 85 23)(13 81 24)(14 82 25)(15 83 21)(26 52 95)(27 53 91)(28 54 92)(29 55 93)(30 51 94)(31 88 77)(32 89 78)(33 90 79)(34 86 80)(35 87 76)(36 111 98)(37 112 99)(38 113 100)(39 114 96)(40 115 97)(46 101 108)(47 102 109)(48 103 110)(49 104 106)(50 105 107)(66 73 117)(67 74 118)(68 75 119)(69 71 120)(70 72 116)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 86)(7 87)(8 88)(9 89)(10 90)(11 117)(12 118)(13 119)(14 120)(15 116)(16 53)(17 54)(18 55)(19 51)(20 52)(21 70)(22 66)(23 67)(24 68)(25 69)(31 44)(32 45)(33 41)(34 42)(35 43)(36 48)(37 49)(38 50)(39 46)(40 47)(56 76)(57 77)(58 78)(59 79)(60 80)(61 95)(62 91)(63 92)(64 93)(65 94)(71 82)(72 83)(73 84)(74 85)(75 81)(96 101)(97 102)(98 103)(99 104)(100 105)(106 112)(107 113)(108 114)(109 115)(110 111)
(1 78)(2 79)(3 80)(4 76)(5 77)(6 54)(7 55)(8 51)(9 52)(10 53)(11 117)(12 118)(13 119)(14 120)(15 116)(16 90)(17 86)(18 87)(19 88)(20 89)(21 72)(22 73)(23 74)(24 75)(25 71)(26 58)(27 59)(28 60)(29 56)(30 57)(31 65)(32 61)(33 62)(34 63)(35 64)(36 110)(37 106)(38 107)(39 108)(40 109)(41 91)(42 92)(43 93)(44 94)(45 95)(46 114)(47 115)(48 111)(49 112)(50 113)(66 84)(67 85)(68 81)(69 82)(70 83)(96 101)(97 102)(98 103)(99 104)(100 105)
(1 58)(2 59)(3 60)(4 56)(5 57)(6 63)(7 64)(8 65)(9 61)(10 62)(11 102)(12 103)(13 104)(14 105)(15 101)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 78)(27 79)(28 80)(29 76)(30 77)(31 51)(32 52)(33 53)(34 54)(35 55)(36 67)(37 68)(38 69)(39 70)(40 66)(71 113)(72 114)(73 115)(74 111)(75 112)(81 106)(82 107)(83 108)(84 109)(85 110)(86 92)(87 93)(88 94)(89 95)(90 91)(96 116)(97 117)(98 118)(99 119)(100 120)
(1 12 78 98)(2 13 79 99)(3 14 80 100)(4 15 76 96)(5 11 77 97)(6 107 54 69)(7 108 55 70)(8 109 51 66)(9 110 52 67)(10 106 53 68)(16 24 90 112)(17 25 86 113)(18 21 87 114)(19 22 88 115)(20 23 89 111)(26 118 58 103)(27 119 59 104)(28 120 60 105)(29 116 56 101)(30 117 57 102)(31 40 65 84)(32 36 61 85)(33 37 62 81)(34 38 63 82)(35 39 64 83)(41 49 91 75)(42 50 92 71)(43 46 93 72)(44 47 94 73)(45 48 95 74)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 6A 6B 6C 6D 6E 10A ··· 10L 10M ··· 10T 10U ··· 10AJ 10AK ··· 10AR 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20P 20Q ··· 20AF 30A ··· 30L 30M ··· 30T 60A ··· 60P order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 6 10 ··· 10 10 ··· 10 10 ··· 10 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 3 3 3 3 6 6 2 2 2 2 2 6 6 6 6 1 1 1 1 2 2 2 4 4 1 ··· 1 2 ··· 2 3 ··· 3 6 ··· 6 4 4 4 4 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4 4 ··· 4

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C10 C20 S3 D4 D6 D6 C4×S3 C5×S3 C5×D4 S3×C10 S3×C10 S3×C20 S3×D4 C5×S3×D4 kernel C5×S3×C22⋊C4 C5×D6⋊C4 C5×C6.D4 C15×C22⋊C4 S3×C2×C20 S3×C22×C10 S3×C2×C10 S3×C22⋊C4 D6⋊C4 C6.D4 C3×C22⋊C4 S3×C2×C4 S3×C23 C22×S3 C5×C22⋊C4 S3×C10 C2×C20 C22×C10 C2×C10 C22⋊C4 D6 C2×C4 C23 C22 C10 C2 # reps 1 2 1 1 2 1 8 4 8 4 4 8 4 32 1 4 2 1 4 4 16 8 4 16 2 8

Matrix representation of C5×S3×C22⋊C4 in GL5(𝔽61)

 20 0 0 0 0 0 20 0 0 0 0 0 20 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 1 60
,
 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 1
,
 50 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 60

G:=sub<GL(5,GF(61))| [20,0,0,0,0,0,20,0,0,0,0,0,20,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,60,60],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[50,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60] >;

C5×S3×C22⋊C4 in GAP, Magma, Sage, TeX

C_5\times S_3\times C_2^2\rtimes C_4
% in TeX

G:=Group("C5xS3xC2^2:C4");
// GroupNames label

G:=SmallGroup(480,759);
// by ID

G=gap.SmallGroup(480,759);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,891,226,15686]);
// Polycyclic

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

׿
×
𝔽