Copied to
clipboard

G = C5×C12.D4order 480 = 25·3·5

Direct product of C5 and C12.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C12.D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C60 — C5×C4.Dic3 — C5×C12.D4
 Lower central C3 — C6 — C2×C6 — C5×C12.D4
 Upper central C1 — C10 — C2×C20 — D4×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, C3, C4, C22, C22, C5, C6, C6, C8, C2×C4, D4, C23, C10, C10, C12, C2×C6, C2×C6, C15, M4(2), C2×D4, C20, C2×C10, C2×C10, C3⋊C8, C2×C12, C3×D4, C22×C6, C30, C30, C4.D4, C40, C2×C20, C5×D4, C22×C10, C4.Dic3, C6×D4, C60, C2×C30, C2×C30, C5×M4(2), D4×C10, C12.D4, C5×C3⋊C8, C2×C60, D4×C15, C22×C30, C5×C4.D4, C5×C4.Dic3, D4×C30, C5×C12.D4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C10, Dic3, D6, C22⋊C4, C20, C2×C10, C2×Dic3, C3⋊D4, C5×S3, C4.D4, C2×C20, C5×D4, C6.D4, C5×Dic3, S3×C10, C5×C22⋊C4, C12.D4, C10×Dic3, C5×C3⋊D4, C5×C4.D4, C5×C6.D4, C5×C12.D4

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

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 12A 12B 15A 15B 15C 15D 20A ··· 20H 30A ··· 30L 30M ··· 30AB 40A ··· 40P 60A ··· 60H order 1 2 2 2 2 3 4 4 5 5 5 5 6 6 6 6 6 6 6 8 8 8 8 10 10 10 10 10 10 10 10 10 ··· 10 12 12 15 15 15 15 20 ··· 20 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 2 4 4 2 2 2 1 1 1 1 2 2 2 4 4 4 4 12 12 12 12 1 1 1 1 2 2 2 2 4 ··· 4 4 4 2 2 2 2 2 ··· 2 2 ··· 2 4 ··· 4 12 ··· 12 4 ··· 4

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + image C1 C2 C2 C4 C5 C10 C10 C20 S3 D4 D6 Dic3 C3⋊D4 C5×S3 C5×D4 S3×C10 C5×Dic3 C5×C3⋊D4 C4.D4 C12.D4 C5×C4.D4 C5×C12.D4 kernel C5×C12.D4 C5×C4.Dic3 D4×C30 C22×C30 C12.D4 C4.Dic3 C6×D4 C22×C6 D4×C10 C60 C2×C20 C22×C10 C20 C2×D4 C12 C2×C4 C23 C4 C15 C5 C3 C1 # reps 1 2 1 4 4 8 4 16 1 2 1 2 4 4 8 4 8 16 1 2 4 8

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

 91 0 0 0 0 91 0 0 0 0 91 0 0 0 0 91
,
 15 227 0 0 101 226 0 0 225 15 0 225 156 15 16 0
,
 240 0 224 0 170 0 240 240 0 1 1 0 0 0 1 0
,
 240 0 224 0 0 0 240 1 0 1 1 0 71 0 1 0
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