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G = C2×D5×C3⋊D4order 480 = 25·3·5

Direct product of C2, D5 and C3⋊D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D5×C3⋊D4, D304C23, C30.50C24, Dic152C23, C65(D4×D5), C307(C2×D4), (C6×D5)⋊18D4, C235(S3×D5), C158(C22×D4), (C2×C30)⋊3C23, D64(C22×D5), (C23×D5)⋊8S3, (C6×D5)⋊8C23, (C22×C10)⋊8D6, (S3×C10)⋊4C23, (C22×D5)⋊16D6, D108(C22×S3), (C22×C6)⋊11D10, C6.50(C23×D5), (C2×Dic3)⋊16D10, (C22×S3)⋊11D10, C157D419C22, C15⋊D419C22, C3⋊D2018C22, C10.50(S3×C23), (C22×C30)⋊7C22, (C5×Dic3)⋊2C23, Dic32(C22×D5), (D5×Dic3)⋊17C22, (C2×Dic15)⋊19C22, (C10×Dic3)⋊15C22, (C22×D15)⋊14C22, C36(C2×D4×D5), C223(C2×S3×D5), (C3×D5)⋊3(C2×D4), C102(C2×C3⋊D4), (D5×C22×C6)⋊5C2, (C22×S3×D5)⋊9C2, C52(C22×C3⋊D4), (C2×D5×Dic3)⋊24C2, (C2×S3×D5)⋊13C22, (D5×C2×C6)⋊19C22, (C2×C6)⋊6(C22×D5), (C2×C157D4)⋊24C2, (C10×C3⋊D4)⋊13C2, (C2×C15⋊D4)⋊22C2, (C2×C3⋊D20)⋊22C2, C2.50(C22×S3×D5), (S3×C2×C10)⋊10C22, (C2×C10)⋊5(C22×S3), (C5×C3⋊D4)⋊14C22, SmallGroup(480,1122)

Series: Derived Chief Lower central Upper central

C1C30 — C2×D5×C3⋊D4
C1C5C15C30C6×D5C2×S3×D5C22×S3×D5 — C2×D5×C3⋊D4
C15C30 — C2×D5×C3⋊D4
C1C22C23

Generators and relations for C2×D5×C3⋊D4
 G = < a,b,c,d,e,f | a2=b5=c2=d3=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: 2556 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], C6, C6 [×2], C6 [×8], C2×C4 [×6], D4 [×16], C23, C23 [×20], D5 [×4], D5 [×4], C10, C10 [×2], C10 [×4], Dic3 [×2], Dic3 [×2], D6 [×2], D6 [×14], C2×C6, C2×C6 [×2], C2×C6 [×20], C15, C22×C4, C2×D4 [×12], C24 [×2], Dic5 [×2], C20 [×2], D10 [×8], D10 [×22], C2×C10, C2×C10 [×2], C2×C10 [×6], C2×Dic3, C2×Dic3 [×5], C3⋊D4 [×4], C3⋊D4 [×12], C22×S3, C22×S3 [×9], C22×C6, C22×C6 [×10], C5×S3 [×2], C3×D5 [×4], C3×D5 [×2], D15 [×2], C30, C30 [×2], C30 [×2], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5 [×2], C22×D5 [×4], C22×D5 [×13], C22×C10, C22×C10, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4 [×11], S3×C23, C23×C6, C5×Dic3 [×2], Dic15 [×2], S3×D5 [×8], C6×D5 [×8], C6×D5 [×10], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C23×D5, C23×D5, C22×C3⋊D4, D5×Dic3 [×4], C15⋊D4 [×4], C3⋊D20 [×4], C10×Dic3, C5×C3⋊D4 [×4], C2×Dic15, C157D4 [×4], C2×S3×D5 [×4], C2×S3×D5 [×4], D5×C2×C6 [×2], D5×C2×C6 [×4], D5×C2×C6 [×4], S3×C2×C10, C22×D15, C22×C30, C2×D4×D5, C2×D5×Dic3, C2×C15⋊D4, C2×C3⋊D20, D5×C3⋊D4 [×8], C10×C3⋊D4, C2×C157D4, C22×S3×D5, D5×C22×C6, C2×D5×C3⋊D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C3⋊D4 [×4], C22×S3 [×7], C22×D4, C22×D5 [×7], C2×C3⋊D4 [×6], S3×C23, S3×D5, D4×D5 [×2], C23×D5, C22×C3⋊D4, C2×S3×D5 [×3], C2×D4×D5, D5×C3⋊D4 [×2], C22×S3×D5, C2×D5×C3⋊D4

Smallest permutation representation of C2×D5×C3⋊D4
On 120 points
Generators in S120
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(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 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(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 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)
(1 109 19 94)(2 110 20 95)(3 106 16 91)(4 107 17 92)(5 108 18 93)(6 111 21 96)(7 112 22 97)(8 113 23 98)(9 114 24 99)(10 115 25 100)(11 116 26 101)(12 117 27 102)(13 118 28 103)(14 119 29 104)(15 120 30 105)(31 76 46 61)(32 77 47 62)(33 78 48 63)(34 79 49 64)(35 80 50 65)(36 81 51 66)(37 82 52 67)(38 83 53 68)(39 84 54 69)(40 85 55 70)(41 86 56 71)(42 87 57 72)(43 88 58 73)(44 89 59 74)(45 90 60 75)
(1 64)(2 65)(3 61)(4 62)(5 63)(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)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O 3 4A4B4C4D5A5B6A···6G6H···6O10A···10F10G10H10I10J10K10L10M10N15A15B20A20B20C20D30A···30N
order122222222222222234444556···66···610···10101010101010101015152020202030···30
size111122555566101030302663030222···210···102···244441212121244121212124···4

72 irreducible representations

dim11111111122222222224444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D5D6D6D10D10D10D10C3⋊D4S3×D5D4×D5C2×S3×D5D5×C3⋊D4
kernelC2×D5×C3⋊D4C2×D5×Dic3C2×C15⋊D4C2×C3⋊D20D5×C3⋊D4C10×C3⋊D4C2×C157D4C22×S3×D5D5×C22×C6C23×D5C6×D5C2×C3⋊D4C22×D5C22×C10C2×Dic3C3⋊D4C22×S3C22×C6D10C23C6C22C2
# reps11118111114261282282468

Matrix representation of C2×D5×C3⋊D4 in GL4(𝔽61) generated by

60000
06000
00600
00060
,
1000
0100
0001
006017
,
1000
0100
0001
0010
,
473300
01300
0010
0001
,
405800
52100
0010
0001
,
403400
52100
00600
00060
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,17],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[47,0,0,0,33,13,0,0,0,0,1,0,0,0,0,1],[40,5,0,0,58,21,0,0,0,0,1,0,0,0,0,1],[40,5,0,0,34,21,0,0,0,0,60,0,0,0,0,60] >;

C2×D5×C3⋊D4 in GAP, Magma, Sage, TeX

C_2\times D_5\times C_3\rtimes D_4
% in TeX

G:=Group("C2xD5xC3:D4");
// GroupNames label

G:=SmallGroup(480,1122);
// by ID

G=gap.SmallGroup(480,1122);
# by ID

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

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

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