Copied to
clipboard

G = C5×C12.46D4order 480 = 25·3·5

Direct product of C5 and C12.46D4

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

Aliases: C5×C12.46D4, C60.224D4, C20.63D12, C4.11(C5×D12), C12.46(C5×D4), (C22×S3).C20, (C2×D12).6C10, (C2×C20).213D6, C159(C4.D4), M4(2)⋊3(C5×S3), (C5×M4(2))⋊7S3, C4.Dic32C10, C22.4(S3×C20), C10.55(D6⋊C4), (C10×D12).16C2, (C3×M4(2))⋊7C10, C20.114(C3⋊D4), C30.97(C22⋊C4), (C15×M4(2))⋊17C2, (C2×C60).342C22, (S3×C2×C10).7C4, C2.9(C5×D6⋊C4), C31(C5×C4.D4), (C2×C6).2(C2×C20), (C2×C4).1(S3×C10), C4.21(C5×C3⋊D4), C6.8(C5×C22⋊C4), (C2×C10).63(C4×S3), (C2×C30).126(C2×C4), (C2×C12).12(C2×C10), (C5×C4.Dic3)⋊14C2, SmallGroup(480,142)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C12.46D4
C1C3C6C2×C6C2×C12C2×C60C10×D12 — C5×C12.46D4
C3C6C2×C6 — C5×C12.46D4
C1C10C2×C20C5×M4(2)

Generators and relations for C5×C12.46D4
 G = < a,b,c,d,e | a5=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=bc, cd=dc, ce=ec, ede=d-1 >

Subgroups: 292 in 92 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], D6 [×4], C2×C6, C15, M4(2), M4(2), C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8, C24, D12 [×2], C2×C12, C22×S3 [×2], C5×S3 [×2], C30, C30, C4.D4, C40 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C4.Dic3, C3×M4(2), C2×D12, C60 [×2], S3×C10 [×4], C2×C30, C5×M4(2), C5×M4(2), D4×C10, C12.46D4, C5×C3⋊C8, C120, C5×D12 [×2], C2×C60, S3×C2×C10 [×2], C5×C4.D4, C5×C4.Dic3, C15×M4(2), C10×D12, C5×C12.46D4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], D6, C22⋊C4, C20 [×2], C2×C10, C4×S3, D12, C3⋊D4, C5×S3, C4.D4, C2×C20, C5×D4 [×2], D6⋊C4, S3×C10, C5×C22⋊C4, C12.46D4, S3×C20, C5×D12, C5×C3⋊D4, C5×C4.D4, C5×D6⋊C4, C5×C12.46D4

Smallest permutation representation of C5×C12.46D4
On 120 points
Generators in S120
(1 58 41 51 24)(2 59 42 52 17)(3 60 43 53 18)(4 61 44 54 19)(5 62 45 55 20)(6 63 46 56 21)(7 64 47 49 22)(8 57 48 50 23)(9 85 110 75 36)(10 86 111 76 37)(11 87 112 77 38)(12 88 105 78 39)(13 81 106 79 40)(14 82 107 80 33)(15 83 108 73 34)(16 84 109 74 35)(25 101 72 92 117)(26 102 65 93 118)(27 103 66 94 119)(28 104 67 95 120)(29 97 68 96 113)(30 98 69 89 114)(31 99 70 90 115)(32 100 71 91 116)
(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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(74 78)(76 80)(82 86)(84 88)(89 93)(91 95)(98 102)(100 104)(105 109)(107 111)(114 118)(116 120)
(1 99 75)(2 100 76)(3 101 77)(4 102 78)(5 103 79)(6 104 80)(7 97 73)(8 98 74)(9 41 90)(10 42 91)(11 43 92)(12 44 93)(13 45 94)(14 46 95)(15 47 96)(16 48 89)(17 32 111)(18 25 112)(19 26 105)(20 27 106)(21 28 107)(22 29 108)(23 30 109)(24 31 110)(33 63 67)(34 64 68)(35 57 69)(36 58 70)(37 59 71)(38 60 72)(39 61 65)(40 62 66)(49 113 83)(50 114 84)(51 115 85)(52 116 86)(53 117 87)(54 118 88)(55 119 81)(56 120 82)
(2 6)(3 7)(9 90)(10 95)(11 96)(12 93)(13 94)(14 91)(15 92)(16 89)(17 21)(18 22)(25 108)(26 105)(27 106)(28 111)(29 112)(30 109)(31 110)(32 107)(33 71)(34 72)(35 69)(36 70)(37 67)(38 68)(39 65)(40 66)(42 46)(43 47)(49 53)(52 56)(59 63)(60 64)(73 101)(74 98)(75 99)(76 104)(77 97)(78 102)(79 103)(80 100)(81 119)(82 116)(83 117)(84 114)(85 115)(86 120)(87 113)(88 118)

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

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

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

105 conjugacy classes

class 1 2A2B2C2D 3 4A4B5A5B5C5D6A6B8A8B8C8D10A10B10C10D10E10F10G10H10I···10P12A12B12C15A15B15C15D20A···20H24A24B24C24D30A30B30C30D30E30F30G30H40A···40H40I···40P60A···60H60I60J60K60L120A···120P
order122223445555668888101010101010101010···101212121515151520···2024242424303030303030303040···4040···4060···6060606060120···120
size11212122221111244412121111222212···1222422222···24444222244444···412···122···244444···4

105 irreducible representations

dim11111111112222222222224444
type++++++++++
imageC1C2C2C2C4C5C10C10C10C20S3D4D6D12C3⋊D4C4×S3C5×S3C5×D4S3×C10C5×D12C5×C3⋊D4S3×C20C4.D4C12.46D4C5×C4.D4C5×C12.46D4
kernelC5×C12.46D4C5×C4.Dic3C15×M4(2)C10×D12S3×C2×C10C12.46D4C4.Dic3C3×M4(2)C2×D12C22×S3C5×M4(2)C60C2×C20C20C20C2×C10M4(2)C12C2×C4C4C4C22C15C5C3C1
# reps111144444161212224848881248

Matrix representation of C5×C12.46D4 in GL4(𝔽241) generated by

98000
09800
00980
00098
,
9081226153
20510612034
198864716
1554329239
,
10125189
01728
002400
000240
,
24011088
240098231
002401
002400
,
1240205127
024016936
001240
000240
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[90,205,198,155,81,106,86,43,226,120,47,29,153,34,16,239],[1,0,0,0,0,1,0,0,125,72,240,0,189,8,0,240],[240,240,0,0,1,0,0,0,10,98,240,240,88,231,1,0],[1,0,0,0,240,240,0,0,205,169,1,0,127,36,240,240] >;

C5×C12.46D4 in GAP, Magma, Sage, TeX

C_5\times C_{12}._{46}D_4
% in TeX

G:=Group("C5xC12.46D4");
// GroupNames label

G:=SmallGroup(480,142);
// by ID

G=gap.SmallGroup(480,142);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,2803,136,2111,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^5,b*d=d*b,e*b*e=b*c,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽