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## G = C2×A4×Dic5order 480 = 25·3·5

### Direct product of C2, A4 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×A4×Dic5
 Chief series C1 — C5 — C2×C10 — C22×C10 — C10×A4 — A4×Dic5 — C2×A4×Dic5
 Lower central C2×C10 — C2×A4×Dic5
 Upper central C1 — C22

Generators and relations for C2×A4×Dic5
G = < a,b,c,d,e,f | a2=b2=c2=d3=e10=1, f2=e5, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=bc=cb, be=eb, bf=fb, dcd-1=b, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 580 in 132 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, C23, C23, C23, C10, C10, C10, C12, A4, C2×C6, C15, C22×C4, C24, Dic5, Dic5, C2×C10, C2×C10, C2×C12, C2×A4, C2×A4, C30, C23×C4, C2×Dic5, C2×Dic5, C22×C10, C22×C10, C22×C10, C4×A4, C22×A4, C3×Dic5, C5×A4, C2×C30, C22×Dic5, C22×Dic5, C23×C10, C2×C4×A4, C6×Dic5, C10×A4, C10×A4, C23×Dic5, A4×Dic5, A4×C2×C10, C2×A4×Dic5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D5, C12, A4, C2×C6, Dic5, D10, C2×C12, C2×A4, C3×D5, C2×Dic5, C4×A4, C22×A4, C3×Dic5, C6×D5, C2×C4×A4, C6×Dic5, D5×A4, A4×Dic5, C2×D5×A4, C2×A4×Dic5

Smallest permutation representation of C2×A4×Dic5
On 120 points
Generators in S120
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 111)(18 112)(19 113)(20 114)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 41)(40 42)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 88)(72 89)(73 90)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(91 108)(92 109)(93 110)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 41)(40 42)(71 88)(72 89)(73 90)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(91 108)(92 109)(93 110)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 111)(18 112)(19 113)(20 114)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 88)(72 89)(73 90)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)
(1 56 36)(2 57 37)(3 58 38)(4 59 39)(5 60 40)(6 51 31)(7 52 32)(8 53 33)(9 54 34)(10 55 35)(11 102 82)(12 103 83)(13 104 84)(14 105 85)(15 106 86)(16 107 87)(17 108 88)(18 109 89)(19 110 90)(20 101 81)(21 68 41)(22 69 42)(23 70 43)(24 61 44)(25 62 45)(26 63 46)(27 64 47)(28 65 48)(29 66 49)(30 67 50)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)
(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 75 6 80)(2 74 7 79)(3 73 8 78)(4 72 9 77)(5 71 10 76)(11 70 16 65)(12 69 17 64)(13 68 18 63)(14 67 19 62)(15 66 20 61)(21 89 26 84)(22 88 27 83)(23 87 28 82)(24 86 29 81)(25 85 30 90)(31 100 36 95)(32 99 37 94)(33 98 38 93)(34 97 39 92)(35 96 40 91)(41 109 46 104)(42 108 47 103)(43 107 48 102)(44 106 49 101)(45 105 50 110)(51 120 56 115)(52 119 57 114)(53 118 58 113)(54 117 59 112)(55 116 60 111)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A ··· 6F 10A ··· 10F 10G ··· 10N 12A ··· 12H 15A 15B 15C 15D 30A ··· 30L order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 6 ··· 6 10 ··· 10 10 ··· 10 12 ··· 12 15 15 15 15 30 ··· 30 size 1 1 1 1 3 3 3 3 4 4 5 5 5 5 15 15 15 15 2 2 4 ··· 4 2 ··· 2 6 ··· 6 20 ··· 20 8 8 8 8 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 6 6 6 type + + + + - + + + + + - + image C1 C2 C2 C3 C4 C6 C6 C12 D5 Dic5 D10 C3×D5 C3×Dic5 C6×D5 A4 C2×A4 C2×A4 C4×A4 D5×A4 A4×Dic5 C2×D5×A4 kernel C2×A4×Dic5 A4×Dic5 A4×C2×C10 C23×Dic5 C10×A4 C22×Dic5 C23×C10 C22×C10 C22×A4 C2×A4 C2×A4 C24 C23 C23 C2×Dic5 Dic5 C2×C10 C10 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 2 4 2 4 8 4 1 2 1 4 2 4 2

Matrix representation of C2×A4×Dic5 in GL7(𝔽61)

 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 0 48 0 60
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 47 60
,
 47 0 0 0 0 0 0 0 47 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 48 47 59 0 0 0 0 60 0 0 0 0 0 0 0 0 13
,
 44 60 0 0 0 0 0 1 0 0 0 0 0 0 0 0 17 1 0 0 0 0 0 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 37 53 0 0 0 0 0 34 24 0 0 0 0 0 0 0 41 34 0 0 0 0 0 8 20 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1

G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,48,0,0,0,0,0,60,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,47,0,0,0,0,0,0,60],[47,0,0,0,0,0,0,0,47,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,48,60,0,0,0,0,0,47,0,0,0,0,0,0,59,0,13],[44,1,0,0,0,0,0,60,0,0,0,0,0,0,0,0,17,60,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[37,34,0,0,0,0,0,53,24,0,0,0,0,0,0,0,41,8,0,0,0,0,0,34,20,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

C2×A4×Dic5 in GAP, Magma, Sage, TeX

C_2\times A_4\times {\rm Dic}_5
% in TeX

G:=Group("C2xA4xDic5");
// GroupNames label

G:=SmallGroup(480,1044);
// by ID

G=gap.SmallGroup(480,1044);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,84,648,271,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^10=1,f^2=e^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=b*c=c*b,b*e=e*b,b*f=f*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽