Copied to
clipboard

G = C5×C12.D4order 480 = 25·3·5

Direct product of C5 and C12.D4

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

Aliases: C5×C12.D4, C60.141D4, C12.8(C5×D4), (C6×D4).2C10, (D4×C10).7S3, (D4×C30).12C2, (C2×C20).215D6, C4.Dic33C10, (C22×C6).2C20, C20.92(C3⋊D4), C1512(C4.D4), (C22×C30).12C4, C23.2(C5×Dic3), (C2×C60).346C22, C22.2(C10×Dic3), (C22×C10).4Dic3, C30.118(C22⋊C4), C10.34(C6.D4), C32(C5×C4.D4), (C2×C4).3(S3×C10), (C2×D4).2(C5×S3), C4.13(C5×C3⋊D4), (C2×C6).28(C2×C20), C6.14(C5×C22⋊C4), (C2×C30).196(C2×C4), (C2×C12).16(C2×C10), (C5×C4.Dic3)⋊15C2, C2.4(C5×C6.D4), (C2×C10).40(C2×Dic3), SmallGroup(480,152)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C12.D4
C1C3C6C2×C6C2×C12C2×C60C5×C4.Dic3 — C5×C12.D4
C3C6C2×C6 — C5×C12.D4
C1C10C2×C20D4×C10

Generators and relations for C5×C12.D4
 G = < a,b,c,d | a5=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b5, dcd-1=b3c3 >

Subgroups: 196 in 92 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], C2×C6, C2×C6 [×4], C15, M4(2) [×2], C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C30, C30 [×3], C4.D4, C40 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C4.Dic3 [×2], C6×D4, C60 [×2], C2×C30, C2×C30 [×4], C5×M4(2) [×2], D4×C10, C12.D4, C5×C3⋊C8 [×2], C2×C60, D4×C15 [×2], C22×C30 [×2], C5×C4.D4, C5×C4.Dic3 [×2], D4×C30, C5×C12.D4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], Dic3 [×2], D6, C22⋊C4, C20 [×2], C2×C10, C2×Dic3, C3⋊D4 [×2], C5×S3, C4.D4, C2×C20, C5×D4 [×2], C6.D4, C5×Dic3 [×2], S3×C10, C5×C22⋊C4, C12.D4, C10×Dic3, C5×C3⋊D4 [×2], C5×C4.D4, C5×C6.D4, C5×C12.D4

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

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

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

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

105 conjugacy classes

class 1 2A2B2C2D 3 4A4B5A5B5C5D6A6B6C6D6E6F6G8A8B8C8D10A10B10C10D10E10F10G10H10I···10P12A12B15A15B15C15D20A···20H30A···30L30M···30AB40A···40P60A···60H
order12222344555566666668888101010101010101010···1012121515151520···2030···3030···3040···4060···60
size112442221111222444412121212111122224···44422222···22···24···412···124···4

105 irreducible representations

dim1111111122222222224444
type++++++-+
imageC1C2C2C4C5C10C10C20S3D4D6Dic3C3⋊D4C5×S3C5×D4S3×C10C5×Dic3C5×C3⋊D4C4.D4C12.D4C5×C4.D4C5×C12.D4
kernelC5×C12.D4C5×C4.Dic3D4×C30C22×C30C12.D4C4.Dic3C6×D4C22×C6D4×C10C60C2×C20C22×C10C20C2×D4C12C2×C4C23C4C15C5C3C1
# reps121448416121244848161248

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

91000
09100
00910
00091
,
1522700
10122600
225150225
15615160
,
24002240
1700240240
0110
0010
,
24002240
002401
0110
71010
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[15,101,225,156,227,226,15,15,0,0,0,16,0,0,225,0],[240,170,0,0,0,0,1,0,224,240,1,1,0,240,0,0],[240,0,0,71,0,0,1,0,224,240,1,1,0,1,0,0] >;

C5×C12.D4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,152);
// by ID

G=gap.SmallGroup(480,152);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,1410,136,4204,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations

׿
×
𝔽