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## G = D5×C4○D12order 480 = 25·3·5

### Direct product of D5 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D5×C4○D12
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C4×S3×D5 — D5×C4○D12
 Lower central C15 — C30 — D5×C4○D12
 Upper central C1 — C4 — C2×C4

Generators and relations for D5×C4○D12
G = < a,b,c,d,e | a5=b2=c4=e2=1, d6=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d5 >

Subgroups: 1628 in 328 conjugacy classes, 112 normal (60 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×4], C6, C6 [×4], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×2], D5 [×3], C10, C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6, C2×C6 [×4], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C2×C10 [×2], Dic6, Dic6 [×3], C4×S3 [×2], C4×S3 [×6], D12, D12 [×3], C2×Dic3 [×2], C3⋊D4 [×2], C3⋊D4 [×6], C2×C12, C2×C12 [×5], C22×S3 [×2], C22×C6, C5×S3 [×2], C3×D5 [×2], C3×D5, D15 [×2], C30, C30, C2×C4○D4, Dic10 [×3], C4×D5 [×4], C4×D5 [×6], D20 [×3], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C22×D5, C22×D5 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12, C4○D12 [×7], C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], D30 [×2], C2×C30, C2×C4×D5, C2×C4×D5 [×2], C4○D20 [×3], D4×D5 [×3], D42D5 [×3], Q8×D5, Q82D5, C5×C4○D4, C2×C4○D12, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], D5×C12 [×4], C6×Dic5, C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, C2×S3×D5 [×2], D5×C2×C6, D5×C4○D4, D5×Dic6, D6.D10 [×2], D125D5, C12.28D10, C4×S3×D5 [×2], D5×D12, Dic3.D10 [×2], D5×C3⋊D4 [×2], D5×C2×C12, C5×C4○D12, D6011C2, D5×C4○D12
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], C4○D12 [×2], S3×C23, S3×D5, C23×D5, C2×C4○D12, C2×S3×D5 [×3], D5×C4○D4, C22×S3×D5, D5×C4○D12

Smallest permutation representation of D5×C4○D12
On 120 points
Generators in S120
(1 105 59 110 25)(2 106 60 111 26)(3 107 49 112 27)(4 108 50 113 28)(5 97 51 114 29)(6 98 52 115 30)(7 99 53 116 31)(8 100 54 117 32)(9 101 55 118 33)(10 102 56 119 34)(11 103 57 120 35)(12 104 58 109 36)(13 47 64 75 94)(14 48 65 76 95)(15 37 66 77 96)(16 38 67 78 85)(17 39 68 79 86)(18 40 69 80 87)(19 41 70 81 88)(20 42 71 82 89)(21 43 72 83 90)(22 44 61 84 91)(23 45 62 73 92)(24 46 63 74 93)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 61)(23 62)(24 63)(73 92)(74 93)(75 94)(76 95)(77 96)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 109)(105 110)(106 111)(107 112)(108 113)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 106 103 100)(98 107 104 101)(99 108 105 102)(109 118 115 112)(110 119 116 113)(111 120 117 114)
(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 87)(2 86)(3 85)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 88)(13 98)(14 97)(15 108)(16 107)(17 106)(18 105)(19 104)(20 103)(21 102)(22 101)(23 100)(24 99)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 84)(34 83)(35 82)(36 81)(37 50)(38 49)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(61 118)(62 117)(63 116)(64 115)(65 114)(66 113)(67 112)(68 111)(69 110)(70 109)(71 120)(72 119)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 10 10 10 10 10 10 10 12 12 12 12 12 12 12 12 15 15 20 20 20 20 20 20 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 2 5 5 6 6 10 30 30 2 1 1 2 5 5 6 6 10 30 30 2 2 2 2 2 10 10 10 10 2 2 4 4 12 12 12 12 2 2 2 2 10 10 10 10 4 4 2 2 2 2 4 4 12 12 12 12 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 C4○D4 D10 D10 D10 D10 D10 C4○D12 S3×D5 C2×S3×D5 C2×S3×D5 D5×C4○D4 D5×C4○D12 kernel D5×C4○D12 D5×Dic6 D6.D10 D12⋊5D5 C12.28D10 C4×S3×D5 D5×D12 Dic3.D10 D5×C3⋊D4 D5×C2×C12 C5×C4○D12 D60⋊11C2 C2×C4×D5 C4○D12 C4×D5 C2×Dic5 C2×C20 C22×D5 C3×D5 Dic6 C4×S3 D12 C3⋊D4 C2×C12 D5 C2×C4 C4 C22 C3 C1 # reps 1 1 2 1 1 2 1 2 2 1 1 1 1 2 4 1 1 1 4 2 4 2 4 2 8 2 4 2 4 8

Matrix representation of D5×C4○D12 in GL4(𝔽61) generated by

 44 1 0 0 16 60 0 0 0 0 1 0 0 0 0 1
,
 60 60 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 11 0 0 0 0 11
,
 60 0 0 0 0 60 0 0 0 0 21 0 0 0 60 32
,
 1 0 0 0 0 1 0 0 0 0 25 30 0 0 28 36
G:=sub<GL(4,GF(61))| [44,16,0,0,1,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,60,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,11,0,0,0,0,11],[60,0,0,0,0,60,0,0,0,0,21,60,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,25,28,0,0,30,36] >;

D5×C4○D12 in GAP, Magma, Sage, TeX

D_5\times C_4\circ D_{12}
% in TeX

G:=Group("D5xC4oD12");
// GroupNames label

G:=SmallGroup(480,1090);
// by ID

G=gap.SmallGroup(480,1090);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,346,1356,18822]);
// Polycyclic

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

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