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G = C6×D4×D5order 480 = 25·3·5

Direct product of C6, D4 and D5

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

Aliases: C6×D4×D5, C607C23, C30.73C24, C102(C6×D4), C20⋊(C22×C6), D207(C2×C6), (D4×C10)⋊5C6, C3014(C2×D4), C234(C6×D5), (C6×D20)⋊27C2, (C2×D20)⋊11C6, (D4×C30)⋊12C2, (C2×C12)⋊29D10, (C2×C30)⋊7C23, (C23×D5)⋊7C6, C127(C22×D5), C1515(C22×D4), (C2×C60)⋊21C22, (C6×D5)⋊11C23, D102(C22×C6), (C22×C6)⋊10D10, C10.5(C23×C6), C6.73(C23×D5), (D5×C12)⋊22C22, (C3×D20)⋊37C22, (D4×C15)⋊29C22, (C3×Dic5)⋊9C23, Dic51(C22×C6), (C22×C30)⋊13C22, (C6×Dic5)⋊35C22, C52(D4×C2×C6), C41(D5×C2×C6), (C2×C4×D5)⋊3C6, (C2×C4)⋊6(C6×D5), C222(D5×C2×C6), (C2×C20)⋊2(C2×C6), (D5×C2×C12)⋊13C2, (C4×D5)⋊3(C2×C6), (C5×D4)⋊5(C2×C6), C5⋊D41(C2×C6), (C2×C5⋊D4)⋊9C6, (C6×C5⋊D4)⋊24C2, C2.6(D5×C22×C6), (D5×C2×C6)⋊22C22, (D5×C22×C6)⋊10C2, (C2×C6)⋊5(C22×D5), (C22×C10)⋊5(C2×C6), (C2×C10)⋊2(C22×C6), (C2×Dic5)⋊8(C2×C6), (C22×D5)⋊7(C2×C6), (C3×C5⋊D4)⋊17C22, SmallGroup(480,1139)

Series: Derived Chief Lower central Upper central

C1C10 — C6×D4×D5
C1C5C10C30C6×D5D5×C2×C6D5×C22×C6 — C6×D4×D5
C5C10 — C6×D4×D5
C1C2×C6C6×D4

Generators and relations for C6×D4×D5
 G = < a,b,c,d,e | a6=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1616 in 472 conjugacy classes, 194 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×4], C22 [×34], C5, C6, C6 [×2], C6 [×12], C2×C4, C2×C4 [×5], D4 [×4], D4 [×12], C23 [×2], C23 [×19], D5 [×4], D5 [×4], C10, C10 [×2], C10 [×4], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×34], C15, C22×C4, C2×D4, C2×D4 [×11], C24 [×2], Dic5 [×2], C20 [×2], D10 [×10], D10 [×20], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C12, C2×C12 [×5], C3×D4 [×4], C3×D4 [×12], C22×C6 [×2], C22×C6 [×19], C3×D5 [×4], C3×D5 [×4], C30, C30 [×2], C30 [×4], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×D5 [×10], C22×D5 [×8], C22×C10 [×2], C22×C12, C6×D4, C6×D4 [×11], C23×C6 [×2], C3×Dic5 [×2], C60 [×2], C6×D5 [×10], C6×D5 [×20], C2×C30, C2×C30 [×4], C2×C30 [×4], C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C23×D5 [×2], D4×C2×C6, D5×C12 [×4], C3×D20 [×4], C6×Dic5, C3×C5⋊D4 [×8], C2×C60, D4×C15 [×4], D5×C2×C6, D5×C2×C6 [×10], D5×C2×C6 [×8], C22×C30 [×2], C2×D4×D5, D5×C2×C12, C6×D20, C3×D4×D5 [×8], C6×C5⋊D4 [×2], D4×C30, D5×C22×C6 [×2], C6×D4×D5
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], D5, C2×C6 [×35], C2×D4 [×6], C24, D10 [×7], C3×D4 [×4], C22×C6 [×15], C3×D5, C22×D4, C22×D5 [×7], C6×D4 [×6], C23×C6, C6×D5 [×7], D4×D5 [×2], C23×D5, D4×C2×C6, D5×C2×C6 [×7], C2×D4×D5, C3×D4×D5 [×2], D5×C22×C6, C6×D4×D5

Smallest permutation representation of C6×D4×D5
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 97 104)(2 62 98 105)(3 63 99 106)(4 64 100 107)(5 65 101 108)(6 66 102 103)(7 109 44 17)(8 110 45 18)(9 111 46 13)(10 112 47 14)(11 113 48 15)(12 114 43 16)(19 51 32 25)(20 52 33 26)(21 53 34 27)(22 54 35 28)(23 49 36 29)(24 50 31 30)(37 58 85 81)(38 59 86 82)(39 60 87 83)(40 55 88 84)(41 56 89 79)(42 57 90 80)(67 119 76 93)(68 120 77 94)(69 115 78 95)(70 116 73 96)(71 117 74 91)(72 118 75 92)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 103)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 51)(20 52)(21 53)(22 54)(23 49)(24 50)(25 32)(26 33)(27 34)(28 35)(29 36)(30 31)(37 81)(38 82)(39 83)(40 84)(41 79)(42 80)(55 88)(56 89)(57 90)(58 85)(59 86)(60 87)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 119)(68 120)(69 115)(70 116)(71 117)(72 118)(73 96)(74 91)(75 92)(76 93)(77 94)(78 95)
(1 95 41 29 109)(2 96 42 30 110)(3 91 37 25 111)(4 92 38 26 112)(5 93 39 27 113)(6 94 40 28 114)(7 104 78 79 36)(8 105 73 80 31)(9 106 74 81 32)(10 107 75 82 33)(11 108 76 83 34)(12 103 77 84 35)(13 99 117 85 51)(14 100 118 86 52)(15 101 119 87 53)(16 102 120 88 54)(17 97 115 89 49)(18 98 116 90 50)(19 46 63 71 58)(20 47 64 72 59)(21 48 65 67 60)(22 43 66 68 55)(23 44 61 69 56)(24 45 62 70 57)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 13)(7 64)(8 65)(9 66)(10 61)(11 62)(12 63)(19 77)(20 78)(21 73)(22 74)(23 75)(24 76)(25 120)(26 115)(27 116)(28 117)(29 118)(30 119)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 88)(38 89)(39 90)(40 85)(41 86)(42 87)(43 106)(44 107)(45 108)(46 103)(47 104)(48 105)(49 92)(50 93)(51 94)(52 95)(53 96)(54 91)(55 81)(56 82)(57 83)(58 84)(59 79)(60 80)(97 112)(98 113)(99 114)(100 109)(101 110)(102 111)

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O3A3B4A4B4C4D5A5B6A···6F6G···6N6O···6V6W···6AD10A···10F10G···10N12A12B12C12D12E12F12G12H15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order1222222222222222334444556···66···66···66···610···1010···101212121212121212151515152020202030···3030···3060···60
size1111222255551010101011221010221···12···25···510···102···24···4222210101010222244442···24···44···4

120 irreducible representations

dim11111111111111222222222244
type+++++++++++++
imageC1C2C2C2C2C2C2C3C6C6C6C6C6C6D4D5D10D10D10C3×D4C3×D5C6×D5C6×D5C6×D5D4×D5C3×D4×D5
kernelC6×D4×D5D5×C2×C12C6×D20C3×D4×D5C6×C5⋊D4D4×C30D5×C22×C6C2×D4×D5C2×C4×D5C2×D20D4×D5C2×C5⋊D4D4×C10C23×D5C6×D5C6×D4C2×C12C3×D4C22×C6D10C2×D4C2×C4D4C23C6C2
# reps1118212222164244228484416848

Matrix representation of C6×D4×D5 in GL4(𝔽61) generated by

60000
06000
00140
00014
,
60000
06000
006053
00461
,
60000
06000
0010
001560
,
0100
601700
0010
0001
,
06000
60000
0010
0001
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,14,0,0,0,0,14],[60,0,0,0,0,60,0,0,0,0,60,46,0,0,53,1],[60,0,0,0,0,60,0,0,0,0,1,15,0,0,0,60],[0,60,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[0,60,0,0,60,0,0,0,0,0,1,0,0,0,0,1] >;

C6×D4×D5 in GAP, Magma, Sage, TeX

C_6\times D_4\times D_5
% in TeX

G:=Group("C6xD4xD5");
// GroupNames label

G:=SmallGroup(480,1139);
// by ID

G=gap.SmallGroup(480,1139);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,409,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽