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G = C5×D4○D12order 480 = 25·3·5

Direct product of C5 and D4○D12

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

Aliases: C5×D4○D12, C30.95C24, C60.242C23, C15142+ 1+4, (C5×D4)⋊30D6, (S3×D4)⋊5C10, D48(S3×C10), (C2×C20)⋊23D6, Q88(S3×C10), (C5×Q8)⋊29D6, C4○D128C10, (C2×D12)⋊13C10, D1211(C2×C10), (C10×D12)⋊29C2, (C2×C60)⋊30C22, Q83S35C10, (S3×C20)⋊15C22, Dic612(C2×C10), C32(C5×2+ 1+4), (C5×D12)⋊41C22, (D4×C15)⋊40C22, C6.12(C23×C10), C10.80(S3×C23), (Q8×C15)⋊35C22, D6.6(C22×C10), (S3×C10).42C23, C20.239(C22×S3), (C2×C30).260C23, C12.26(C22×C10), (C5×Dic6)⋊39C22, Dic3.8(C22×C10), (C5×Dic3).44C23, (C5×S3×D4)⋊12C2, (C2×C4)⋊4(S3×C10), C4○D45(C5×S3), C4.26(S3×C2×C10), (C4×S3)⋊2(C2×C10), (C2×C12)⋊5(C2×C10), (C3×C4○D4)⋊4C10, (C5×C4○D4)⋊12S3, (C3×D4)⋊9(C2×C10), C3⋊D45(C2×C10), (C3×Q8)⋊8(C2×C10), C22.4(S3×C2×C10), (C5×C4○D12)⋊18C2, (C15×C4○D4)⋊14C2, (S3×C2×C10)⋊16C22, C2.13(S3×C22×C10), (C22×S3)⋊4(C2×C10), (C5×Q83S3)⋊12C2, (C5×C3⋊D4)⋊21C22, (C2×C6).4(C22×C10), (C2×C10).23(C22×S3), SmallGroup(480,1161)

Series: Derived Chief Lower central Upper central

C1C6 — C5×D4○D12
C1C3C6C30S3×C10S3×C2×C10C5×S3×D4 — C5×D4○D12
C3C6 — C5×D4○D12
C1C10C5×C4○D4

Generators and relations for C5×D4○D12
 G = < a,b,c,d,e | a5=b4=c2=e2=1, d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >

Subgroups: 788 in 332 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C5, S3 [×6], C6, C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×15], Q8, Q8, C23 [×6], C10, C10 [×9], Dic3 [×2], C12, C12 [×3], D6 [×6], D6 [×6], C2×C6 [×3], C15, C2×D4 [×9], C4○D4, C4○D4 [×5], C20, C20 [×3], C20 [×2], C2×C10 [×3], C2×C10 [×12], Dic6, C4×S3 [×6], D12 [×9], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], C5×S3 [×6], C30, C30 [×3], 2+ 1+4, C2×C20 [×3], C2×C20 [×6], C5×D4 [×3], C5×D4 [×15], C5×Q8, C5×Q8, C22×C10 [×6], C2×D12 [×3], C4○D12 [×3], S3×D4 [×6], Q83S3 [×2], C3×C4○D4, C5×Dic3 [×2], C60, C60 [×3], S3×C10 [×6], S3×C10 [×6], C2×C30 [×3], D4×C10 [×9], C5×C4○D4, C5×C4○D4 [×5], D4○D12, C5×Dic6, S3×C20 [×6], C5×D12 [×9], C5×C3⋊D4 [×6], C2×C60 [×3], D4×C15 [×3], Q8×C15, S3×C2×C10 [×6], C5×2+ 1+4, C10×D12 [×3], C5×C4○D12 [×3], C5×S3×D4 [×6], C5×Q83S3 [×2], C15×C4○D4, C5×D4○D12
Quotients: C1, C2 [×15], C22 [×35], C5, S3, C23 [×15], C10 [×15], D6 [×7], C24, C2×C10 [×35], C22×S3 [×7], C5×S3, 2+ 1+4, C22×C10 [×15], S3×C23, S3×C10 [×7], C23×C10, D4○D12, S3×C2×C10 [×7], C5×2+ 1+4, S3×C22×C10, C5×D4○D12

Smallest permutation representation of C5×D4○D12
On 120 points
Generators in S120
(1 64 43 59 16)(2 65 44 60 17)(3 66 45 49 18)(4 67 46 50 19)(5 68 47 51 20)(6 69 48 52 21)(7 70 37 53 22)(8 71 38 54 23)(9 72 39 55 24)(10 61 40 56 13)(11 62 41 57 14)(12 63 42 58 15)(25 100 109 81 90)(26 101 110 82 91)(27 102 111 83 92)(28 103 112 84 93)(29 104 113 73 94)(30 105 114 74 95)(31 106 115 75 96)(32 107 116 76 85)(33 108 117 77 86)(34 97 118 78 87)(35 98 119 79 88)(36 99 120 80 89)
(1 88 7 94)(2 89 8 95)(3 90 9 96)(4 91 10 85)(5 92 11 86)(6 93 12 87)(13 76 19 82)(14 77 20 83)(15 78 21 84)(16 79 22 73)(17 80 23 74)(18 81 24 75)(25 72 31 66)(26 61 32 67)(27 62 33 68)(28 63 34 69)(29 64 35 70)(30 65 36 71)(37 104 43 98)(38 105 44 99)(39 106 45 100)(40 107 46 101)(41 108 47 102)(42 97 48 103)(49 109 55 115)(50 110 56 116)(51 111 57 117)(52 112 58 118)(53 113 59 119)(54 114 60 120)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 103)(98 104)(99 105)(100 106)(101 107)(102 108)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 66)(62 65)(63 64)(67 72)(68 71)(69 70)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 90)(86 89)(87 88)(91 96)(92 95)(93 94)(97 98)(99 108)(100 107)(101 106)(102 105)(103 104)(109 116)(110 115)(111 114)(112 113)(117 120)(118 119)

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

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

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

135 conjugacy classes

class 1 2A2B2C2D2E···2J 3 4A4B4C4D4E4F5A5B5C5D6A6B6C6D10A10B10C10D10E···10P10Q···10AN12A12B12C12D12E15A15B15C15D20A···20P20Q···20X30A30B30C30D30E···30P60A···60H60I···60T
order122222···23444444555566661010101010···1010···1012121212121515151520···2020···203030303030···3060···6060···60
size112226···622222661111244411112···26···62244422222···26···622224···42···24···4

135 irreducible representations

dim111111111111222222224444
type++++++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D6D6D6C5×S3S3×C10S3×C10S3×C102+ 1+4D4○D12C5×2+ 1+4C5×D4○D12
kernelC5×D4○D12C10×D12C5×C4○D12C5×S3×D4C5×Q83S3C15×C4○D4D4○D12C2×D12C4○D12S3×D4Q83S3C3×C4○D4C5×C4○D4C2×C20C5×D4C5×Q8C4○D4C2×C4D4Q8C15C5C3C1
# reps13362141212248413314121241248

Matrix representation of C5×D4○D12 in GL4(𝔽61) generated by

34000
03400
00340
00034
,
1020
0102
600600
060060
,
1000
0100
600600
060060
,
462300
382300
004623
003823
,
462300
381500
004623
003815
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,34,0,0,0,0,34],[1,0,60,0,0,1,0,60,2,0,60,0,0,2,0,60],[1,0,60,0,0,1,0,60,0,0,60,0,0,0,0,60],[46,38,0,0,23,23,0,0,0,0,46,38,0,0,23,23],[46,38,0,0,23,15,0,0,0,0,46,38,0,0,23,15] >;

C5×D4○D12 in GAP, Magma, Sage, TeX

C_5\times D_4\circ D_{12}
% in TeX

G:=Group("C5xD4oD12");
// GroupNames label

G:=SmallGroup(480,1161);
// by ID

G=gap.SmallGroup(480,1161);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,891,2467,304,15686]);
// Polycyclic

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

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