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## 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 [×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 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

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