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## G = C6×D4×D5order 480 = 25·3·5

### Direct product of C6, D4 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C6×D4×D5
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — D5×C22×C6 — C6×D4×D5
 Lower central C5 — C10 — C6×D4×D5
 Upper central C1 — C2×C6 — C6×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, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, D4, C23, C23, D5, D5, C10, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, C22×C6, C3×D5, C3×D5, C30, C30, C30, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C22×C12, C6×D4, C6×D4, C23×C6, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C2×C30, C2×C30, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, D4×C2×C6, D5×C12, C3×D20, C6×Dic5, C3×C5⋊D4, C2×C60, D4×C15, D5×C2×C6, D5×C2×C6, D5×C2×C6, C22×C30, C2×D4×D5, D5×C2×C12, C6×D20, C3×D4×D5, C6×C5⋊D4, D4×C30, D5×C22×C6, C6×D4×D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C24, D10, C3×D4, C22×C6, C3×D5, C22×D4, C22×D5, C6×D4, C23×C6, C6×D5, D4×D5, C23×D5, D4×C2×C6, D5×C2×C6, C2×D4×D5, C3×D4×D5, 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 53 104)(2 62 54 105)(3 63 49 106)(4 64 50 107)(5 65 51 108)(6 66 52 103)(7 39 60 87)(8 40 55 88)(9 41 56 89)(10 42 57 90)(11 37 58 85)(12 38 59 86)(13 46 70 116)(14 47 71 117)(15 48 72 118)(16 43 67 119)(17 44 68 120)(18 45 69 115)(19 101 32 25)(20 102 33 26)(21 97 34 27)(22 98 35 28)(23 99 36 29)(24 100 31 30)(73 79 111 96)(74 80 112 91)(75 81 113 92)(76 82 114 93)(77 83 109 94)(78 84 110 95)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 103)(7 39)(8 40)(9 41)(10 42)(11 37)(12 38)(13 46)(14 47)(15 48)(16 43)(17 44)(18 45)(19 101)(20 102)(21 97)(22 98)(23 99)(24 100)(25 32)(26 33)(27 34)(28 35)(29 36)(30 31)(49 63)(50 64)(51 65)(52 66)(53 61)(54 62)(55 88)(56 89)(57 90)(58 85)(59 86)(60 87)(67 119)(68 120)(69 115)(70 116)(71 117)(72 118)(73 96)(74 91)(75 92)(76 93)(77 94)(78 95)(79 111)(80 112)(81 113)(82 114)(83 109)(84 110)
(1 45 41 29 109)(2 46 42 30 110)(3 47 37 25 111)(4 48 38 26 112)(5 43 39 27 113)(6 44 40 28 114)(7 34 81 108 16)(8 35 82 103 17)(9 36 83 104 18)(10 31 84 105 13)(11 32 79 106 14)(12 33 80 107 15)(19 96 63 71 58)(20 91 64 72 59)(21 92 65 67 60)(22 93 66 68 55)(23 94 61 69 56)(24 95 62 70 57)(49 117 85 101 73)(50 118 86 102 74)(51 119 87 97 75)(52 120 88 98 76)(53 115 89 99 77)(54 116 90 100 78)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 73)(7 57)(8 58)(9 59)(10 60)(11 55)(12 56)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)(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)(43 100)(44 101)(45 102)(46 97)(47 98)(48 99)(49 114)(50 109)(51 110)(52 111)(53 112)(54 113)(61 80)(62 81)(63 82)(64 83)(65 84)(66 79)(91 104)(92 105)(93 106)(94 107)(95 108)(96 103)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3A 3B 4A 4B 4C 4D 5A 5B 6A ··· 6F 6G ··· 6N 6O ··· 6V 6W ··· 6AD 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 4 4 4 4 5 5 6 ··· 6 6 ··· 6 6 ··· 6 6 ··· 6 10 ··· 10 10 ··· 10 12 12 12 12 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 2 2 5 5 5 5 10 10 10 10 1 1 2 2 10 10 2 2 1 ··· 1 2 ··· 2 5 ··· 5 10 ··· 10 2 ··· 2 4 ··· 4 2 2 2 2 10 10 10 10 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

120 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 C2 C3 C6 C6 C6 C6 C6 C6 D4 D5 D10 D10 D10 C3×D4 C3×D5 C6×D5 C6×D5 C6×D5 D4×D5 C3×D4×D5 kernel C6×D4×D5 D5×C2×C12 C6×D20 C3×D4×D5 C6×C5⋊D4 D4×C30 D5×C22×C6 C2×D4×D5 C2×C4×D5 C2×D20 D4×D5 C2×C5⋊D4 D4×C10 C23×D5 C6×D5 C6×D4 C2×C12 C3×D4 C22×C6 D10 C2×D4 C2×C4 D4 C23 C6 C2 # reps 1 1 1 8 2 1 2 2 2 2 16 4 2 4 4 2 2 8 4 8 4 4 16 8 4 8

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

 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 53 0 0 46 1
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 15 60
,
 0 1 0 0 60 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
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